Yes. Yes to an awful lot of questions you're asking.
Yes, it's too much to expect that six weeks is going to give you what you need to control a horse.
Yes, it's not right of the people at the barn to pressure you into going into a show.
Yes, it's both you and the horse.
Yes, you should probably stick with the horse.
Yes, you are probably over-thinking a lot of this stuff.
Yes, it's probably unnecessary for anyone to hit the horse while tacking it up.
Yes, if you prefer the private lessons, it is your right to ask (and pay) for them instead of a group lesson. Also yes, you may be in a position to learn something more from a group lesson than a private one.
Yes, it is possible to post the trot without stirrups. Yes, there are schools of thought that would consider posting without stirrups to be MORE difficult than posting with them. Yes, you get incredibly valuable things by riding without stirrups and will probably be a better rider in the long run if you can master those skills.
The reason that a horse is successful in a lesson program is because it is able to tune out the vast amounts of totally incorrect stuff that new riders do (including a bunch of stuff that new riders don't even realize they're doing), and respond only to the correct cues. Another reason that the horse is successful in a lesson program is because it's very unlikely to take off and do alarming things. A good lesson horse, for beginners, is going to require both a correct and a relatively assertive cue...a weak cue might just as easily be something the rider isn't intending to ask, so you don't really want the horse to respond to it.
It takes time, effort, and LOTS of practice to get the cues down properly. You are speaking to your horse with your hands, with your butt, with your reins, with where you put your weight, and with your voice. All of those things need to be delivering the same message (like "please trot") at the same time. The horse knows that language. You don't. And while it's possible to a certain degree to learn that language from a book or a video, it's just the same as any language - when you start trying to actually hold a conversation, you realize just how useful the book or tape is NOT.
The horse may be lazy. I think it is generally considered advisable to put a beginner (and with 6 weeks, you are definitely in that category) on a lazier more mellow horse than on an energetic hot one. Riding a lazy horse means that you can focus on getting YOUR signals to the horse correct, instead of focusing on keeping the horse from running off with you at the same time you're trying to take on a bunch of information about what signals to send, how, when, and why.
When the horse started to move off as you mounted, that was a sign. That horse might just as well have unfurled a 10-foot banner that said "I KNOW YOU ARE A ROOKIE AND I AM TAKING CHARGE". It's like in a war movie, where some green officer right out of college with zero practical experience shows up to command troops, and the NCO who is "under" that office realizes that if the troops follow the commands that officer is giving, everyone is going to die, so the NCO takes charge.
The horses, in a way, are trusting you with their lives. They're at the bottom of the food chain. They eat only plants. They're designed with hair-trigger reflexes and the ability to run really really fast...because they have to, if they want to live to see another day. Think what it means for something like that to submit to all that tack and a rider. It means they can't run that fast, and they're squashing some of those hair-trigger reflexes. If they're going to do that, they have to believe that you will be able to keep them from getting killed and eaten. I know this is a huge simplification, but in general principles, it's there. When people here talk about getting the horse's "respect" it has to do with demonstrating to that horse that YOU are a leader it can trust to take care of IT and that means demonstrating that you are really on the ball, and doing so in a way that horses can understand.
It sounds like this horse does not see you in this light, nor should it, really...you're like the rookie officer! You haven't seen and done enough, from the horse perspective, to be experienced enough to lead. The goal, then, is for you to learn how to lead, and to demonstrate this consistently to the horse. Then, and only then, will it "respect" you.
This is not a one-shot, either. If you watch horses with each other, you see that this situation (who exactly is going to be in charge here) gets sorted out over, and over, and over. They test each other constantly. It's for their *safety* that they do this, it's a sort of biological imperative. You can expect, as a rider, to get tested over, and over, and over too. It's just part of being a horse person. They don't usually mind losing, just as long as they KNOW.
This would be the case no matter what horse you were riding, and it would be even more the case if you had a more responsive mount. If you had a VERY responsive mount (and the skills to ride that horse, which takes a while to develop) you would be getting tested constantly.
There is no substitute for time spent WITH the horse, and ON the horse.
As far as getting the bridle on, the best way I've found to deal with it - because my horse IS highly responsive, and after a year together and a year of ultra-consistency, he only tests me once a week or so...but when he does, he tends to throw his head in the air, so I've had to deal with that bridle issue before...
There's a trick to it that I tried to describe here, and then realized that would be more confusing than just getting someone at the barn (OTHER than the person who whacked the horse in the face) to show you. IF you are holding the bridle properly as you go to put it on, it is difficult for the horse to raise his head way up, and your left hand is also in a good position to stick your finger into the side of his mouth and get him to open it up for the bit (if that is the problem). It should take someone at the barn no more than 5 minutes to show you how to do it, then it's up to you to practice doing it properly every single time you tack up.