I had no idea they offered milkshakes to go, I'd be very interested to hear the answer! PS- if you are a milkshake lover, the Primetime Cafe and the Plaza Restaurant have great milkshakes!
Don't know for sure. Best to just ask when you are there.
WDW restaurants don't seem to like to do "to go" for walk ups.
They are set up for it as they allow doggie bags.
But at Nine Dragons I could not even get a Chai ice tea to go. I was told they are not set up for it. (Heck, their counter service is right next door and probably share the same kitchen.)
It used to be easy to take out from WDW restaurants. That changed with the dining plan and free dining. Restaurants are much busier now and don't want to be bothered with take-out -- especially since people might try to use TS credits with take-out. Disney prefers that if you don't have time to, or don't want to, actually sit down and use your TS credis that you forfeit them. Also, they have more control over who is using the credits if you actually go in and sit down (as opposed to taking the meal out).
There was so much that was nicer before DDP. Oh well.