Why did you let him in or to do work without you there or having terms already?

I would have that conversation real quick with him (record it on phone or have a witness there if possible) and bring up the fact he did that work and purchase without your authorization and you are not going to pay for install or equipment.

If you do not put a red line down, contractors will run right over you. If he gets pissy, just handle it in small claims court. You will pay around $50-100 to sue him and if he doesn't show up to court you get a default judgement. If he does, have him provide a signed agreement for the work. Either way, sounds like it would be a win for you. (I've never sued anyone, but if they tried to screw me, I'd screw them twice as hard)

Now as for the AC it looks like you already bought.... i can't help you there. I exhaust as much heat and bring in fresh air. (I have around 3800ish cfm exhaust)