About animal speed and hunting, I'd like to point you to
UnReal World gameplay.
It's a game somewhat like H&H (except it's solo).
There are 2 ways to hunt there :
- passive : using traps and baits (there are various. depending of the game you are willing to catch).
- active : you basically sneak onto an animal try to hit it (easier with ranged weapons like bows / spears / ...).
If you do hit it, and depending on your skills, and some luck, it'll have different effect.
For example the animal can get injured and run slower, or bleed, or fall unconscious...
Then it's all about following it, closely enough so he gets tired and you can deliver the final blow.
Animals leave tracks which you can follow more or less easily (depending on a skill).
About running speed, an Elk will be WAY faster than you. Don't even expect to get up close as you would with HnH's deers for instance.
Though if he's injured that's another story.
H&H hunting system is way too "arcade" and quite broken Imo.
Animals can't break signs fences when fleeing... twitch hunting... OP river ranged attacks...
Any experimented player can kill a bear without too much hassle after a few hours playing on a brand new character.
Go tickle a bear on URW without dogs and have fun re-rolling.

So yeah, hunting is too easy. Rabbits, foxes and deers should at least flee when they see a human and their aura should be augmented at least to the double (maybe depending on the player stealth).