What happens when your horse dies? Does it disappear entirely, or is there somewhere you can go to still 'visit' it?

You can buy a memorial pasture and keep them there otherwise they will go to the rainbow bridge.

Ok, thank you

The same place where you get barns and upgrades. Buy a pasture and convert it into a memorial garden :)

Grow>Upgrades>Pastures

As long as a horse has living descendents in the game it will still be visitable, even if it is not in a memorial garden.

Only horses without living descendents will get deleted totally.

I see. Ok, thank you