Not in my case ... I've had goats and horses. The goats eat branches, leaves, bark, etc. That's about it. My family's horses (mustangs) have ate/licked/nibbled on:
soap (nice teeth marks)
their fencing (completely chewed off the wood)
a dog rug
a frying pan
a leather tractor seat (it's now completely gone).
You don't; not strictly. You could say "mitt skjerf", but if you use the other form, you put "skjerfet mitt". The second form is more common in casual conversation, the first form is more "academic". I read it in my head as "the scarf of mine" vs "my scarf", if you need help remembering when to do which.