On the show, when Food “won”, was Adam allowed to doggie bag leftovers?
I don’t know that he would want to. The schedule was pretty tight and he didn’t want to fill up before traveling to the next city. Plus where would be store it?
Nor do I think the production assistants probably wanted the half-eaten steak.
Excellent point yellow.
You haven’t worked with productions crews much I see. 
There’s usually a couple guys who would fight off a hyena for half a steak.
Forget the leftovers from the challenge. They usually made several examples of either the challenge dish or some other things the place is known for, and Adam would maybe take a bite from one of them. I hope the crew got to eat the rest.
“Allowed” to? Most of these challenges are along the lines of “it’s free if you finish it, otherwise free.” So presumably it would be paid for, or maybe more likely the restaurant comps it due to free publicity. I can’t see why he (they) couldn’t have taken it “home,” if wanted. Whether they did or not is a separate issue, and maybe best answered in an interview or commentary (I don’t know of any).
Lurking: your last sentence is why I asked. 