How do I get my animals to not congregate at the same place my dwarfs do?
Just make a bunch of ropes and chains and tie them up. They breed via spores, as long as they're not caged they can be chained at opposite ends of the map and still reproduce. Your FPS will thank you.
Another solution would be to build cages(b to build then j for cage) and assign the animals to the cages(query the built cage and assign). A built cage can have any number of tame animals stuffed into it. Newborn animals will appear outside the cages, so you may have to occasionally assign the new animals to the cage as they show up. If you're concerned about FPS cages should be pretty good for that, caged animals won't pathfind at all.