Where do we receive notification when a horse levels up to a 10. It is annoying as it means that my Barn Manager cannot enter them for competitions until I notice

Its not too hard to find out with a single look at the barn if a horse leveled up to 10. And you can look at the level up report, too.

They do not show on the level up report

You don't have too many horses, just skim through your barns on 'Training View' and look for 10s! :)