Assuming Michigan and Pennsylvania run their local shows about the same, then you would be best to direct most of these questions to the show grounds you're interested in. Every show association or individual show is run a little bit differently and follows a different set of rules. Some are more lenient than others.
Where I've been showing my mare lately, it's a local riding club that holds a series of shows throughout the season. There are English and Western classes. English is in the morning and the Western classes start after lunch. They also hold fun shows with gaming and jumping classes. For these shows, competition is fairly stiff but the rules are relaxed. For instance, you could enter any class wearing any tack or clothing, and as long as it's not deemed unsafe, you would not be dismissed from the class, but most of the competitors do compete in traditional tack and formal show attire and these well-presented horse and riders will likely place higher.
I don't know a whole lot about western showing but I don't see any reason why bitless would be illegal, as long as you are using a legitimate, safe bitless bridle, and not something unusually harsh or complicated or some homemade, jerry-rigged contraption.
You do NOT need a trainer to show. I show without my trainer all the time.
Generally, you will need proof of negative coggins (current within the past year) and your horse's vaccination records. Most show grounds require at least a current rabies vaccination, but as long as your horse is vaccinated with all the recommended vaccines for your area, you will be fine. Some clubs or grounds require a health certificate as well. Definitely call the show officials and make sure you have all the required paperwork before going to the show. Usually, shows only require you give your name and contact information and your horse's name. Then they give you a back number. You should NOT need your horse's pedigree or registration information if it is not a breed affiliated show.
Some shows charge a day fee. You pay one flat fee and enter as many classes as you want. The riding club I rode with this past year charged by the class. It was $7 per class. The cost would depend on the show, so this is something you will have to call and ask about.
You do not have to stay on the grounds. If the show is only one day, then everybody will trailer in in the morning, spot clean, tack up and warm up, and then be ready and present for their classes. Once they are done showing for the day, they untack, load up and leave. If the show goes on for multiple days, you are welcome to rent a stall, or just trailer in and work out of your trailer for the days and times you're currently showing. You can talk to the show association or grounds to find out what their specific rules are regarding arrival times and parking.