When your box is full, it's full. Meaning you won't be able to
place more pokemon there if you don't release or move pokemon from
that box first to make room. And no, you don't get more boxes when
you fill one, you have to buy a new one from Bill. Alternatively,
you can pay Bill to extend the capacity of your already existing
boxes. Whichever works better for you.
Do you mean Route 53? I'm not sure what 'missions' you mean so I
can't really help you with that. I can help you with the miltank
and combee though. I have some spares so I'll put up a private
trade for you with one of each to get you started. :]