The siren call of guide writing has lured me yet again. Here's the culmination of my findings and learnings about animal breeding over the past three worlds bundled up in a nice little package for everyone. Don't like my rambling and just want the meat of the guide? Check out the "Keep It Simple Sevenless", KISS for short, section I made specially for you

Anyone is free to post this guide elsewhere, or translated it, so long as a link to the original is included. If there are edits, please credit them to the writer where it's changed.
These names are long, I use them a lot. Here's the shortforms I use
Q = quality
BQ = breeding quality
Q% = percentage quality trait
KISS
The version of the guide for people who don't want to listen to me ramble. It says what to do, but doesn't cover the mechanics or why you should do it
Feeding

Honestly, the KISS section covers this in all the detail needed. It's a trade off between space and sanity. Given the option I think going for hemp/wheat/barley option is best, just ignore the stuff that gets dropped on the ground unless you need the byproduct. Definitely not worth the effort for the extra bit of fodder from straw. Since we're here though, I'll repeat what's in the KISS section. Also, obviously, these numbers assume you have the Farmer credo.
Assumptions: Turnips 1 harvest per day, hemp/wheat/barley 1 harvest per 3 days, pumpkin 1 harvest per 7 days. This can vary based on speed nodes, adjust to your location
Turnips: Insane yield/tile of farmland, must harvest them within a timed window every day. Beehives not needed, but will make wax/honey. Will drain sanity.
Hemp/Wheat/Barley: Don't require beehives which is good for 10,000 tile fodder fields. Need 3x more field space than turnips.
Pumpkins: They produce the most fodder/harvest action. Fiddly to do without bots, Needs beehives, Needs 2x more field space than turnips.
2000 tiles of turnips per 50 unherded breeders or 70 herded breeders. Adjust to the crop you want, also fed the milkers, riding horses, and 20 coops of chickens.
Don't fall into the trap of sticking with turnips past the first month. They will burn you out because if you miss a window they turn into a non-seed crop that destroys your soul to replant. Usually best to just dump it on the ground and replant from saved seeds. Make sure you bust your butt on expansions so you can swap crop types (farm fields don't have to be flat, and you can terraform under growing crops without damaging them). I likewise hate pumpkins because they're fiddly. Ultimately if you can't bot fodder it's better to be realistic and try to either super focus on one trait to breed + some milkers or to only have Q/BQ herds to reduce fodder consumption.
Get herder asap! Reduced food consumption and faster generation time for better quality gain is great!
Breeding

Breeding livestock has long been an interest of mine in haven, my primary occupation for the past 3 worlds. For the sake of being optimal in gains, I've been running 100-150 breeders (fodder field farming without bots mostly because I'm too lazy to write that bot ironically). With herder bonus, we're talking processing/slaughtering as many as 40 animals per day. Eventually my sanity erodes and I give up on the process and shift over to maintenance mode where breeding is mostly stopped or relegated to a "slaughter pen" where you just ignore quality entirely.
Here's the kicker. I've been doing it wrong. I realized in my second world that I could improve gains if I narrowed the focus on each species down. If I only raise cows for milk I go from caring about 7 traits down to 4 (Q, BQ, MilkQuant, Milk Q%). This improved my gains notably, but sorting 40 animals for 4 qualities each day was still very mentally taxing. I came up with a rough algorithm to score the animals to simplify my work where each stat was worth certain points so long as it didn't fall below a threshold (any animal with BQ<Q or MilkQuant <10 was immediately killed).
I had the right idea, but I missed this important change to breeding. Specifically:
Implemented Potjeh's breeding idea. Instead of baby animals using an average of their parents' qualities when determining their own stats, they now select the value from one or the other parent, before adding a bit of a random roll as usual. The objective being to create more selective breeding.
So the key here is that we could breed a cow with Q100 to a cow with Q10, and the offspring will be either Q10+/- OR Q100+/- It will no longer be Q50+/- I say "no longer" like this update wasn't 4 years ago... moving on:
If we extrapolate this concept though, this means we could breed a cow with high Q/BQ with a cow that has high Milk Q%. I'm ignoring Milk Quantity because if we're talking actual meaningful quality gain on animal breeding you should have the fodder to feed a large herd of milking cows since they're much cheaper than breeding cows. You can replace milk quantity with cow quantity in this case. The reason why we want to do this is simple: If we're only trying to roll 1-2 traits positively, we will get a much stronger upward momentum. Bad rolls on other stats don't matter for us, but for someone trying to raise everything evenly it would ruin the breeder. Like gaining 20 quality but losing 5% milk quality. For us we don't care about the milk Q%, we have another herd for that. So now that we've covered why this gets us good rolls, we need to talk about how we combine them into our super cows:
To make a Q/BQ/Q% hybrid, we have a 12.5% chance (50% chance to get the correct trait, for 3 different traits so 0.5*0.5*0.5=0.125 or 12.5%) to get a super cow when breeding together the Q/BQ and the Q% herds. Keep breeding your Q% herd, and every extra 50% Q% or so mix in females from the Q% herd into the B/BQ herd. Look for the lucky super cows, and try to replace your entire Q/BQ breeds with super cows. Once all the breeders have the good Q%, go back to ignoring Q% and breeding just for Q/BQ again.
So the longterm strategy for animals is to have a herd for Quality/Breeding Quality, and then a Quality % herd for each of the other traits you want to chase. Quantity stats can be ignored because you can compensate that with volume. You might ask: Why mix into the Q/BQ herd instead of mixing into the Q% herd? If you're just picking for Q%, BQ dropping can quickly do damage to your Q as well by male BQ softcapping Q. Since it has a lasting multigenerational form of damage, it's less severe than just ignoring the Q%. Puts a pickle on the situation if you want to raise multiple traits for a single animal unless you have multiple Q herds, but that's up to you to decide.
Note1: In Q/BQ herds Males must have higher or equal BQ to their Q. Females you select for their highest trait either Q or BQ, don't balance them. Unbalanced females that match the male stat (100/100 male with 100/50 female for example) mean instead of needing to clone the male (1/4 chance) you only need to get one of his traits for good possible rolls (1/2 chance).
Note2: Although it's unlikely, BQ does actually impact things like truffle snout (and presumably Q% to the tune of BQ has to be higher than Q% - 100) if your BQ is lower than that. Because BQ is -5 to +20, this basically never happens, but I rolled badly 7 generations in a row and a 13BQ male ended up with my best truffle snout. Loe and Behold, all of his piglets were garbage for 2 generations before I realized what was going on. I realized this after writing the guide, and have tried to edit any references that suggest otherwise.
Traits

The issue here is sanity. Each new trait you want to chase involves more breeders and more sorting. Sorting with this method *is* much easier. Your milk% cows? Just sort by milk% and kill everything below a threshold. Done. But ultimately feeding and slaughtering still takes a lot of time. I think even with this shorter method it was easy for me to spend 1+ hr slaughtering/butchering per day. Remember that every animal needs a quality herd on top of this, and if there's nothing starred for an animal I think just having a Q/BQ herd is plenty fine. You can always crossbreed back to a fresh tamed animal if your quality %s manage to drop crazy low.
Note: I included the possibly useful traits and starred the traits I think are most important
Cows
*Milk - The cheeses made from cowmilk are godtier strength and charisma foods. The secondary cheeses are nice variety if you have enough milk/rennet. Midnight Blue cheese can be sold for tokens mid world if that interests you.
Meat - Beef makes some great CON recipes
Pigs
*Meat - Pork is incredible for CON2 and DEX. Pork bollocks are a nutty STR2 powerfood as well.
Quantity - Pork is really good as mentioned, to the point that it's viable to ignore quality and just make a LOT of pork instead. I vaguely remember there being a cap on how much they could produce, but 30 meat quantity vs 10 meat quantity is a pretty huge difference in meat produced already. This is "lazier" but produces pork in much greater quantities very quickly for your village. Might be worth doing instead of meat Q%.
*Hide - Hide quality also determines bone quality. If we're looking at this from a deepworld viewpoint, eventually animals can eclipse all other sources of bones because % quality scales infinitely. Why pigs? Other than making a sick boar tusk helmet, pigs get a bonus to having twins/triplets. This means that they will gain quality faster than other species by having more rolls/week/breeder. That said, it will take at least 6-7 months to see useful bones (250+) and it will likely take 12 months to beat natural bone quality. This trait is something only deep worlders should bother with.
*Truffle nose - Truffles boost a food by a %, and then take some of that boost and convert it into a specific stat. Any pig can find black truffles that give WIL, but with 40+ nose you start finding white truffles which convert the boosted stat into STR. Sadly they are local pooled, so your mileage may vary. Truffle nose doesn't interact with Q at all, so these can be raised solo without ever crossbreeding.
Special note on pigs: Since they don't give suckling maws, let them mature to adult before slaughtering. Spitroasting seems to ignore age though, so you can spitroast them as babies if you want.
Other special note: I didn't cover why I ballparked 6-12 months. You complete a generation once every ~1.5 weeks (~3.5 day pregnancy with herder, 7 days maturing). The maximum you can gain on Q% herds is 5 per gen, and the max you can gain on Q is 20. For the sake of realism, lets assum you get 50% of those gains. So roughly 10Q and 2.5% per week. Assuming you really get this rolling 1 month into the world. After 5 months of this, it's reasonable to say you'll have ~200Q and ~150% herds. Your 200Q might be held up somewhat by fodder quality since it seems easier to get animal quality than crop quality this world for most things. This lands us at Q300 bones, replacing natural sources for most non-faction villages. At 12 months into the world we can have animals Q440 base, but I'm extremely doubtful foddering those will be reasonable. Our Q% will be up to 210% at this point, giving us a maximum theoretical bone of 924. I'm not sure if natural bones that high are possible, but if they were the supply would be so small it'd be meaningless compared to the amount a herd of animals could produce.
Goats
Wool - Asides from generic wool uses, you can make a Pointed Cap that is a very good hat for gilding. Not critical, but some people like them.
Milk - Creates a great dex cheese, but dex needs mostly cap at a certain point. Pork can easily get you to that point, no need to bother with cheese really.
Sheep
Wool - Generic wool uses, and none of the sheep wool specific crafts are meta/good enough to be worth focusing on. Just having a quality herd and ignoring % for either goats or sheep to have some wool is fine.
Milk - Oscypki is the other cheese frequently bought for tokens midworld. Sheep produce half the milk cows do though, so it can be a real pain to produce.
Meat - The main reason to care about their meat is shepherd's pie, a very tedious but powerful recipe for STR2/Con or Con or PER. There are many variations due to how many ingredients go into this. Still not worth the effort in my opinion, but you can make your own decision on that one.
Horses
Stamina - Horses are a bit unique in that you mostly care about stamina (rate of ponypower regenerates). Since they have 3 riding traits, and the other two traits are fine if you just keep them above base stats. Your aim is to make a horse that can maintain a 3 speed indefinitely (I think it's 30 stamina?). Endurance lets you sprint longer, metabolism gives you a bigger "gas tank" since it gives you more pony power per hunger. The amount of hunger a horse has is static, which is why the conversion matters. Overall I'd rate their importance as Stamina>Metabolism>Endurance.
Horse Note: Horses that are ridden to having less than 90% of their hunger bar (the carrot bar) will start taking starvation ticks. This only damages their quality, but if the starvation bar (brown bar when inspecting) hits 0 they will die instantly and dismount their rider. Multiple horses and trading them out frequently during the day is the best way to avoid starvation damage, but ultimately expect them to accumulate it over time and prepare to breed replacements. I prefer to use castrated males for riders so there's no accidental pregnancies that starts consuming more fodder than needed.
Misc note: The BQ of the male will lower the quality of any babies he fathers if his BQ is below their relevant quality. Usually this only matters for Q, but very rarely other traits like truffle snout can wind up capped if you completely ignore it.
Misc note2: Fodder quality acts as a hard limit to the Q an animal can breed at, but you can recover the fodder quality eaten over time by feeding them better food. Starvation (failing to find any fodder to eat) deals permanent damage to the Q of an animal. Q is the only trait these mechanics interact with, all other traits ignore Fodder Q entirely. A half starved horse still works fine for endurance/stamina traits for example.
Managing herds
This is a preference thing, but here's how I manage my herds.
Animals are kept below ground in a minehole directly under the fodder fields. This saves space aboveground, and makes dropping fodder off more convenient. It is critically important that the area underground that you have walled in is under a walled area on the surface. Otherwise there's a chance raiders notice and can minehole into your animal area. Proper defense in haven means mitigating every form of risk, even ones that seem unlikely to you.
Each herd has a name, nothing fancy. My quality herds all start with Q. Males are numbered in the order they should be used in. So Q M3 is better than Q M2. Sometimes you'll have multiple "better than current breeding male" calves growing up, so it's good to know at a glance which is better.
Since you will probably have at least 10x more females than males, how you number them doesn't matter; you'll have to use the animal ledger to sort them. Q F would be fine, but I tend to number them based off their father. So a female born from Q M2 would be Q F3 (since she's the next generation).
I always colour males green to help them stand out, and the females I colour either yellow or light blue and do all of the same herd that colour. Those are the only two colours other than green I find easily readable. Animals intended for slaughter I tag as red, mostly applicable to pigs since you want to let them grow up to maturity before slaughtering since they don't give suckling maws.
Unbranded animals get their name, but don't get a colour until they're branded.
Unmemorized babies are ones that need to be checked
So what does it look like?

I've separated out most of my breeding males to put things on pause since my activity levels dropped and I don't have a fodder bot to feed everything.
Milking stalls
Because animals won't walk through open gates anymore, and we can guide animals specifically, a very efficient and easy to make milking stall system is possible

These gates are made doing this setup, with the blocking bits and posts removed for easier navigation.

The gates are left open since livestock won't walk through them, and faced inwards so it's a bit more compact visually. Replace a stall wall with a feeding trough where needed.
P.S. Apparently males will cross open gates for breeding. Don't keep your milkers near an open pen with males in it I guess?
Anyway that wraps this up for me. Possible I might include taming/coop sections in the future, no promises. Also I'm definitely putting off slaughtering and I just noticed I have multiple breeders around so I gotta cull the weaker ones >.>