What are Angoras and Silkies?

Angora Rabbits are a breed of rabbit that grows long, fine hair that can be used in yarn and textiles, frequently referred to as "angora wool". Depending on the breed, angora rabbits either shed their hair periodically or must be sheared. The wool is second only to cashmere in fineness and is something of a luxury good.

Silkie Chickens are a breed of fancy chicken with long, fluffy feathers that look and feel almost like fur. They come in a variety of colors, but the most well known variety is white. They're regarded as being more of an ornamental chicken than a practical eggs-and-meat chicken, but they are considered something of a delicacy in parts of Asia.
How do we get them?
Angoras and Silkies are purely domestic breeds that are never found in the wild, and this should be reflected in game. You will never, NEVER find an Angora rabbit hopping around in a forest or a flock of Silkie chickens in a meadow. They can only be born in coops and hutches, and any random chick or bunny rabbit has a very small, let's say 1%, chance of being born an Angora or Silkie. The difference will be apparent at birth, and there will be different sprites for an Angora bunny rabbit or a Silkie chick.
Once the bunny or chick grows up, they can breed with other rabbits and chickens, and this will let you get even more Angoras and Silkies. We could go the simple route and say that one special rabbit/chicken breeding with one regular rabbit/chicken gives 50% special and 50% normal offspring, while two special parents gives 100% special offspring. Alternatively, we could go the Mendelian route, and when an Angora rabbit breeds with a normal rabbit, the offspring are all normal, but carry Angora genes. Angora x Carrier = 50% angora + 50% carrier; Normal x Carrier = 50% normal + 50% carrier; Carrier x Carrier = 25% normal + 50% carrier + 25% Angora. In either case, if both parents are Angora or Silkie, all offspring are Angora or Silkie.
Other than the rules governing Angora/Silkie genetics, breeding for special rabbits and chickens works the same way as breeding normal rabbits and chickens. There is no change regarding gestation/incubation time, litter size, or the rules governing offspring quality.
What do we do with them?
These special rabbits and chickens give special products, namely Angora wool and Silkie feathers. Angora wool is a new variety of wool. 2 Angora wool can be crafted into Angora yarn, 3 Angora yarn can be crafted into Angora wool cloth. This should have its own texture when used in things that require cloth. I'm picturing a silvery platinum color. Angora wool can also be used in any recipe that requires regular wool or mohair, like gauze or wool stuffing, but since it's so hard to get, it's best if you get Angora wool out of your inventory before you start crafting with other wool (more on that in a bit). Silkie feathers are a new type of fine feather, but a type that can be farmed for quality. They're a curio that can be studied, probably with LP/H similar to swan feathers, and are usable in any recipe requiring feathers or fine feathers.
As for collecting these special products, I can see two different approaches. Maybe Angoras and Silkies should shed their wool or feathers for you to just pick up from the hutch or coop, or maybe Angoras have to be sheared like a sheep to get wool, and Silkie feathers can only be plucked off of a dead Silkie the same way you get regular chicken feathers. In either case, Angoras aren't a good way of getting a large amount of wool unless you have like 12 rabbit hutches. They give like 1 unit of wool per rabbit per week. Sorry, but it is a luxury good. Even silk is easier to get in bulk, but angora wool is less annoying.
Unique recipes
Why would you want Angora wool or Silkie feathers more than regular wool or feathers? I'm glad you asked! Here are some suggestions for unique crafting recipes:
Luxury Coat (gear) = 5 rabbit hides + 10 angora wool + 5 silk thread. Gives +5 boost to Charisma at base 10 quality, and has high gilding affinities for charisma and agility. Basically looks like a massive and incredibly expensive mink coat.
Silkie boa (gear) = 20 silkie feathers + 2 silk thread. High gilding affinities for charisma and psyche.
Bunny Poff (gilding) = 4 angora wool + rabbit hide + bone glue. Gilds for agility and charisma.
Silkie brooch (gilding) = 5 silkie feathers + kyanite. Very good gilding for Psyche.
Easter bunny (curio) = 3 Easter eggs + 5 angora wool + 2 rabbit hides + 2 string + 4 straw. A little rabbit doll holding a straw basket of Easter eggs. High mental weight, high study time, high LP/hour, and every ingredient can be farmed for quality. But its best feature is that it's adorable. It should be displayable or something.
Silkie fan (curio) = 5 silkie feathers + stone + string. Appearance, mental weight and study time are similar to the feather fan, but the LP/hour is significantly upgraded. It's also much more fluffy, so clearly it's far superior to the regular feather fan. Also makes a great wall decoration!
Other
* Angora rabbits do not run away from Hearthlings or wild animals when dropped in the overworld. They're domesticated, after all.
* Wild animals like foxes react to Angora rabbits and Silkie chickens the same way they react to normal rabbits and chickens, i.e. killing them. So, yeah, don't drop your special rabbits and chickens in the wild.
* Angoras and Silkies don't despawn when dropped in the overworld. I'll confess: the only reason I want this is so you can drop one in your house and have your own pet fluffy bunny or fluffy chicken.
* When slaughtered, Angoras and Silkies give the same products as normal rabbits and chickens (with the exception of Silkie feathers when plucked). You get no additional wool from an Angora rabbit when you slaughter or flay it. When you flay it, you just get a rabbit hide, and when you butcher it, you just get rabbit meat. Dead Angoras and Silkies have their own sprites, but once flayed or plucked, they just become Flayed Rabbit/Plucked Chicken.
* Maybe if we're going to farm these animals for special products, we should be able to breed them for better wool/hide/feather/egg production, the same as you can do with domestic sheep and cows. This might be complicated to implement, though, since rabbits and chickens don't already have those kinds of breeding mechanics.
* Question for readers: should Angoras and Silkies breed any more slowly than regular rabbits/chickens? Basically, once you have a breeding population of Silkies, should there still be a point in keeping regular chickens? Personally I think that getting them in the first place should be difficult and random enough that once you finally get them, you might as well replace your whole existing stock of rabbits/chickens and enjoy the special products in addition to all the same normal products. However, you could also have it so that Silkies lay eggs at about half the rate of normal chickens (which is true of Silkies IRL) so that normal chickens are better to keep for egg and meat production while Silkies are kept just for Silkie feathers. What sounds like a better approach to you?