I know damage assessment was supposed to show what damage you did in more detail as you leveled it but we currently just see it all as if it was 100 for everyone. (Example with 0 skill it used to say you did damage to walking dead, with 40 it said you struck walking dead for some damage, 80 you pierced walking dead for 10-30 damage, 100 what we have now)
Anatomy I think boosts how much healing you get from bandages.
If I remember correctly, you learn the skill by applying bandages to yourself and then Damage Assessment will follow once Anatomy is trained up to a certain level.
Above is correct - you can learn by action, although if you want to buy the book Anatomy Encyclopedia is 2.5g from the Common Librarian in any town. The equivalent skill for boosting the bandage healing for pets is Veterinary.
If you spam bandages to 50 anatomy and you unlock damage assessment, any damage you deal will level the assessment and the parent skill of anatomy. No need to spam bandages anymore