Keep in mind that I'm a junior member, so I'll defer to the opinion of the more senior folks if they disagree.
1. I don't think you're cheating. As long as you're using the proper ammunition (i.e. if 12ga, then no more than 1 1/8oz) and not violating the range rules (my range does not allow "tactical" or short barrel weapons to be used), the gun you use makes no difference. I've seen shooters use O/U, auto-cycling, and even pump action.
2. Your performance might increase if you switch out the full choke for a cylinder or skeet choke. Since skeet is shot at such short ranges, you want your pattern to spread out as quickly as possible to increase your chances of a dead bird.
3. If you're really interested, check out http://www.nssa-nsca.org/
to see if there are any registered events or clubs in your area. I got introduced to skeet about four months ago and fell hard for it. I'm now out each weekend practicing and started entering the registered shoots a month ago. You compete with people at your own skill level for prizes...not against the AAA shooters. I actually took runner up in my class for two events at the club championships here.
4. If you really get into this, you'll find most serious shooters are using an O/U shotgun with a good amount of auto-loaders in there also. The Remington 1100 sporting is a popular (and generally well regarded) choice. Otherwise you'll see most folks shooting Krieghoff, Kolar, Blaser, or Browning (all in O/U). Granted...most of those cost an arm and a leg. I'm saving up over the course of a few years to eventually get myself a top tier competition gun. In the meantime, I shoot with a used Winchester sporting model. Got my first 25-straight a couple weeks back, so it's not like you need the big name guns to break-em-all.
Welcome to the fold of skeet fanatics. I love the hobby, not just for the sport of it, but because the people I've met at my local range have been fantastic. I've found the skeet shooting community to be quite enjoyable.
Correction: From section 1 - When I say the gun makes no difference, I mean in regards to the cheating concern. Having the right gun does make all the difference for shooting and improving your scores. I recommend something with an adjustable comb so that you can tailor the fit to your specific needs.