To answer your first question, not to us but I won't ride like 90 percent of the rides in the park. ok maybe 75. I'm here for the food, theming and spending time with my family. Therefore I'm the pack mule. The cups get stashed in a backpack that's waterproof so if they tip it just gets rinsed and hung to dry in the hotel room.
To your second question, On those rides I will ride (Men in Black for example) the cups have to go into a locker. The lockers aren't tall enough to set the cups upright so you're either dumping or forcing someone else to stand by and hold while you ride. We try to time our beverage refills so we're either stashing empty cups in the locker or I'm holding cups while waiting for my kids to get off Hulk.