Does anyone know when you get the bredding barn is if AI owners make offers to breed with your horses?

Not that I know of.

The only thing you need to watch for is a bug. For what ever reason one of your horses will randomly go to auction - usually from your stable but very occasionally from your breeding barn.
It's easy to check for IF your game has the bug - you just have to check the mid week auctions...just scroll down the reserve prices and if you see $0 - that is your horse.
So you will have to buy it back or let it go.
The sort of positive thing about buying it back - for example you bid 1 million for your horse maybe only 100k comes out of your bank.