Question 3. Can you use the same horse for an entire game?

No.  Polo ponies run the equivalent of one to two miles during a seven-and-a-half-minute chukker, so they must be rested frequently.  At the high-goal level, players ideally will have a fresh horse every period although many will “double” on their best ponies.  Even at this level, it’s rare but not unheard of for a pony to be played three chukkers in the same game—it happened in the famed International Matches between England and the United States.