OK, that actually does help quite a bit!
All right, where to begin? Let's just start out 1 at a time.
1) Did anyone at PetSmart explain the nitrogen cycle to you? I'm going to guess that they didn't, so I would recommend reading this article here:
http://www.aquariumadvice.com/showquestion.php?faq=2&fldAuto=21
The fact that you have both ammonia and nitrites registering means that your nitrogen cycle is in process, which is good, but brings me to #2.
2) Both your ammonia AND nitrite levels are WAY too high! Those levels are almost certain to be causing the fish stress and are only a couple notches from being toxic. I would recommend doing a 25% water change right away and then again every day if necessary until these levels come down. Ideally, you want your ammonia to be 0, nitrite 0, and nitrates up a little bit. This WILL happen with time, but at this point, if you just let nature run its course, you may lose several and possibly all of those fish!
3) You are already a tad overstocked for a 5 gallon. A basic idea of how many fish to keep is 1 inch of fish for each gallon. You probably already have 8 inches of fish, and as all of those tetras will grow to around 2", eventually as those fish mature, will grow to around 16 inches of fish! This also accounts for the severe rise in your ammonia and nitrite levels because the tank may already be overstocked BEFORE the cycle completed.
4) If it IS ich, letting nature take its course will mean all of the fish will be infested and eventually die. Here is another great article to read:
http://www.aquariumadvice.com/showquestion.php?faq=2&fldAuto=32
There are excellent pictures and descriptions there to look at. If it is ich, you will have to treat it one of the ways listed, or it will just get worse. It sounds like your water temp is fine for the fish without a heater, but not having a heater will eliminate treating ich with the high heat method. I'm sure someone else with more experience treating it will pop in with more advice if you do determine it is ich.
But anyway, I don't think your mixture of fish is bad. I think they will be fine together. Right now, your most important pressing issues are to lower those levels of ammonia and nitrite through daily 25% water changes if necessary, and to determine for sure whether or not that fish has ich and act quickly if it does! Don't go too crazy with your water change, don't change it all out at once or anything like that, as that will be VERY stressful for the fish. When you change, make sure to get the temp as close as you possibly can, even using a thermometer if necessary. Don't forget chlorine remover.
I know this might all seem overwhelming right now, but trust me, once you get the hang of this, it will get much easier and seem like second nature!
Good luck, and keep us posted!
Paul