1) Animals won't breed if there are already 5 others within 20 meters, so if their roaming area is limited then so is the number they can breed up to.

2) If you effectively have them on an independent island that has proper anti-spawn coverage then, yes, they should be safe.

3) You don't need to put them in cages, but you should eventually put them indoors. At a certain point in the game, you'll start getting raids by flying creatures that will ignore the waters separating your farm island and can completely kill all of your animals before the raid is finished. Barn will need to be completely enclosed as said flying creatures can get in from any small hole, including chimneys.