the A/C does not blow cold, cannot hear the compressor kick in?

It's possible that the connector to the clutch got knocked off and was missed. If you look down next to the A/C compressor it should be obvious if it's hanging loose. If you want to test it without digging you could go to the relay box under the hood and pull the A/C relay out then try putting a jumper between terminals 1 and 2 and listen for the clutch to click, if it does the go to the A/C fuse B17 in the same fuse box and see if it is blown. Or take it back to the shop and see what they say or do. If you are worried the shop might be dishonest you could do the above tests and not say anything if you found it was a blown fuse or connector off. If they come to you and say that they tested it, and the compressor is bad, you know not to trust them.