My purchase isn't showing up
When you purchase an avatar, backgrounds and other DLC, they aren't automatically applied. They become available to use, but you need to go pick them. You'll find most of those items under "options".
Another thing to keep in mind, is that items that you purchase on one name don't show up on another name. So be sure to be sign in with the name you made the purchase with, to access that item.
Where to find items:
Fooms / Trinkets: those are in your foom context menu when you select another player. (with the exception of Joy/no joy which only apply to your own character"
Avatars: options->Avatar
Environment backgrounds: Options->Appearance
Card Deck faces and backs: Options->AppearanceDice and checkers: Options->Appearance
Backgammon AI: Game options when starting a game.