Huh. Sounds like it is time to look for a new LFS to replace the two who seem to be selling you a bill of goods. You should have been advised to get a CUC before introducing your first fish and a Star and Stripe Puffer is not exactly a good fish for a newly cycled tank. Although theoretically omnivorous, it leans more toward carnivore than herbivore. This can worsen your ammonia problem.
As an aside, there are several things to learn from your LFS interaction. First, I never trust a fish store to tell me when my tank is "ready" for fish. They are in the business of selling fish, so they have an incentive to tell you your tank is ready when it may not be. Second, I would not take any advice from a fish seller who doesn't know the difference between a dog face and a star and stripe puffer. My teenage daughter saw a dog face in an aquarium at the LFS and said, "Hey, that fish has a face that looks like a dog's!" Your LFS should be able to tell. Finally, this shows the importance of doing your own research on a potential fish purchase. A reputable store will let you put down a deposit on a fish, holding it for you while you go home and do as much research as you can on that species.
As for skimmers keeping ammnonia down, yes, they can by removing some of the waste from your water before it is broken down into ammonia, nitrites, and nitrates. They do not, however, remove ammonia per se from the water.
In regards to the Ammonia Alert, the purpose of one of these is to quickly see if you have ammonia in your tank. Since you know you have ammonia, I would expend the effort to check it with a test kit as opposed to watching the Ammonia Alert.
It (fortunately) sounds like your live rock is cycled, having been taken out of a friend's tank. However, if it was kept out of water for any significant length of time, some die-off would occur.
+1 on having a quarantine tank. You worked hard to get your tank set up. No use messing it up by potentially introducing a disease causing organism to your DT.
Sorry for the long post. Hope this helps a little.