Get a premium TF2 account that’s able to trade if I receive an old gift copy

I would check again following today's update.

Also, try visiting the Mann Co. store once, even if you aren't buying anything.

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.