Does buying the Orange Box grant you premium status in TF2?

Yes, it does. Buying (or being gifted) the Orange Box is one of the ways to get "premium status" and the "Proof of Purchase" hat in TF2. In fact, this is one of the original ways to do so.

I own the Orange Box, which I bought a long time ago, and I have both of these things, even though I've never made a purchase via the Mann Co. store. Others on the Steam Forums have reported that even after the game went F2P, this is still the case.

Yes, it grants you premium status. This is why TF2 is still listed as being part of the Orange Box.

However, from what I've heard, it no longer grants the Proof of Purchase if purchased after the date TF2 went Free to Play. People who bought it before that date and have a giftable copy can still gift it to someone to give them the hat.

In my case I have been granted with Proof of Purchase, however haven't got the premium abilities, like trading, more slots etc.