No support. You can easily unfreeze your horse with a couple clicks rather its in a tab or your whole account and it's easy enough to refreeze since there's no cooldown between freezes. Not worth the dev time when there's much better things they could be working on. 

Also no support. There is no cooldown timer and I believe one of my mares I wanted to get an egg from, had been frozen and I forgot that I had froze her and as long as a horse remains frozen, that timer from when you physically bred them, does not go down and if it's a mare? That's a painfully long week made longer if frozen and temporarily forget about you had bred them before freezing like I tend to do.

This is why I try to wait until horses, regardless of their level, to 17/18 and try to get eggs/straws then. Granted, I do have horses over the age of 20 because I have used them but that was before prices were lowered for expanding the breeding clinic storage amount.