There is no such button.
No. You can't buy a HOP on Howrse without opening it. It is impossible.
Horn of Plenty
though HOP's (Horn of Plenty) which can be found in the "Black Market" on Howrse, open it in the luck of finding one. NOTE: You won't automatically find the gem horse, Howrse will tell you on the News Feed when they're in the HOP's. also, you can win them in contests Howrse does every month!
A Horn of Plenty is specifically a "surprise bundle", and the contents are 100% random so there is no way of "getting something good". If you are not happy with the contents of the HoP you open, just spend the pass on the item you were hoping for.
It means to modify your presentation.
it is under the breeding tab
Click on them quickly with your mouse to catch a surprise.
Yes, some of the objectives do give HOP when they are completed.
You click a button
It depends on where on Howrse.com you are looking.
The game saves itself, there is no save button.
You need a purchase transaction code, which can be found on your receipt.