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So, for example, you can shoot from a bow and immediately bite with a wyvern? Can you attack in any combination: as the rider of both, and as the animal of both?
Yes, you can.
What about shooting, casting spells, and movement? Can't you shoot or cast spells? And what about throwing pumpkins and flasks with bombs? Can you do that simultaneously with movement?
You can't move and shoot/cast spells at the same time, but that doesn't apply to animals. Throwing flasks and pumpkins doesn't even hinder movement on your own two feet.
Do animal attacks (hydra, wyvern) go without an accuracy check?
If accuracy isn't specified somewhere, the default value is 50%.
If the effect is applied by the animal, then how does it work?
Based on the rider's spirit.
One more point I haven't described — the minimum accuracy after all other calculations is 5%. That is, there should always remain a 5% chance to hit, even if there's 200% evasion.