Question 1 - no. question 2 - yes.
The trick is to figure out what todo with your mug after our meal - if you are planning to head tot he parks after eating. You either have to carry the mug, or biring it back to the room.
I also find we have to wash it somehow.
For those reasons, we often don't use the mugs much. Soem folks like to carry small bits of sponge preloaded with dish soap. WDW has sinks near the mug refill stations, but no soap. In the room, we sometimes use the 'liquid shower gel' as dish soap. (or shampoo)