HGG Community Forums

I can answer your first question. :P
There's a little arrow next to your horse's name on the barn list and the personal horse's page. That will take you to a page where you can choose from a few different shows. It doesn't really matter which one you take, though. Any horse can be shown twice in a week, but no more. It's nice and simple. :)
If you're upgraded to Basic, you can click a button at the bottom of a barn's page (besides the other options) to Enter All Horses in that barn.
If you're upgraded to Premium member status, horses will be shown automatically. I'm not sure if this means that everyone in your barns at Sunday 12:10 am precisely will be shown and any others must be shown by hand or by barn (I think there are Advanced settings), or if everyone will be shown as soon as they're created or bought and so on.

Rose--with premium upgrade you can set when each barn (except the primary barn which still has to be shown by hand or with the enter this barn button) is shown by picking days. It seems to enter everyone at roll over to that day, so at midnightish game time. However if there aren't enough classes or if you add horses to the barn after that designated show time, those horses don't get shown for that turn, so I go through my barns close to the end of the week and hand enter anyone who got missed!