yeah they do. I've seen lots of people walking around with a tray of boxed food. Just ask when you tell them your order. Room keeping will pick up your tray for you, or you could be nice and take it back.
we always took the food back to our room when we ate there- I am not sure we got it in to go boxes- but whatever remained- trays/ silverware/ plates ? mousekeeping just picked up.. we would go back to the hotel in the middle of the day and often ate there while we rested- and the room was just more comfortable for that
Yep, I think I've done this at least once at every Disney resort I've stayed at. They have plastic covers that snap over the plates so you don't have to worry about spilling.
Just tell them when you order that you'd like it to go and they'll fix it up for you.