I realize that it's not the LBS's fault that your tire was punctured by a screw, but it sounds to me like they're the ones that screwed up on the chain. I think you should take your bike back to them and have them replace it. If they have any integrity at all, they'll replace it at no cost, even if you try to pay for the new chain/link (which is what I would try to do).
I was have shifting problems with my bike 1+ months after a tune-up and I took it to the LBS for an adjustment and they apologized profusely and fixed it for free and I didn't even have to ask!
Good luck! I'm sorry all that happened..