How often can I expect egg-laying animals to lay eggs?
Normal egg-layers lay eggs roughly once per season.
However, and this is the point you're having problems with:
War animals never lay eggs; or, if they are not egglayers, never breed.
I've never had occasion to find out if animals find a nest box while on a chain, but I suppose they should.
But again; captured war animals will not breed.
Ever.