The ribbon race is where you and your partner hold a piece of "ribbon" usually paper between you and go through the gaits announced. The point is to be the last pair without a ripped "ribbon".
Here is what I found:
Musical Chairs – When music stops riders dismount and run to the center with
Horse and sits in chair. Person without chair is out.
I didn't find exactly what Cups was but somethings it might be...
Mint Julep - Riders will hold a cup of colored water, and be asked to walk/trot/canter depending on division. Rider with most Mint Julep left wins.
Champagne Glass - Rider given a champagne glass filled with fluid at one end of arena and must ride back to other end of arena with fluid still in the glass. Fastest time with most fluid in glass wins.
Beer Can or Soda Can Stack - Riders race from a starting line to a barrel and stop long enough to place one beer or soda can on the top of an already stacked pile of cans. Rider places a can so that it stays and then turn around the barrel and races back to finish line. Pile of cans are the same number for each entrant to be fair so an attendant must be present to adjust the stack for each entrant.