I was thinking of making a Free Hand Fighter from the APG. Is it possible to go with Two Weapon Fighting and Improved Unarmed Strike (for the free hand) as an option for this archtype? The way I'm reading the rules, this should be OK, and still be able to use the benefits of this archtype.
I think you would be able to do that. There is nothing restriction you from striking with your free hand when not using one of the Archetype Abilities.
I never considered it this way before, but now that you mention it, I think it would be pretty fun. You could focus on the Two-Weapon Fighting and Unarmed Strike feat chains, and get into your Dex build as well. Could be very effective.
Also, I don't think there is anything saying you can't kick with that unarmed strike either. It gives the ole "Aha!" of a swashbuckler giving the bad guys a kick down the stairs while blocking a blow with his rapier.