! I'd love to answer some of these questions, and welcome to HF! Keep an open mind and you'll learn a lot from people here!
1) Topline is the muscle from the withers to the croup on the horse. It is their core, the equivelant of our abdominals for us (though the back is not their "abs," they do have abdominals as well, but their core is their back and topline). A horse with the spinal processes may have weight issues, and yes usually a horse with a weak topline struggles to maintain strength in their back and it is easier for them to become sore. But genetically, some horses will never have the topline a quarter horse has. Arabs and Thoroughbreds tend to have A frame backs, and you can get the topline better but the back will never be flat for 51% of these horses. The solution to a horse with a weak topline is to ride them back to front, your instructor can tell you more of what that is, to engage the topline muscles and target-train them.
2) Crops are tools. They are not evil. You are actually doing more damage to a horse by kicking and nagging than using a crop. Horses like "one and done" method of training, so one smack with a crop when they aren't accepting the leg is all they will need, ideally, for the entire ride. I ride in spurs and a dressage whip, and I frequently use the whip to remind my mare to go off my natural aids, not my artificial (crop) aids. Do not discount crops, for kicking and squeezing and getting nothing is far, far more harmful than using an artificial aid to help instill respect to the aids in your horse. Used correctly and without the intention to cause pain, they are harmless.
3) You don't need to feel bad about shopping for barns. First, you need to find a place that aligns with your values and goals in riding. Not every barn is for every person. If the trainer gets mad at you for trying other barns, that tells me they are insecure about their training abilities, and that's a red flag.
4) It's perfectly fine to pet all the horses, just don't feed them. The reason is not because you are breaching any etiquette, but there may be horses with nutritional sensitivities and you don't know how much "sugar" could create a reaction. It is for the safety of the horse, and has nothing to do with trying to control you. Myself, I don't mind if my mare is fed treats, but we have a highly sugar sensitive horse and they are not allowed treats at all, as one treat could set off a laminitic episode.
Be nice, be courteous, and (because I am assuming you are young) most of all, don't be a know-it-all. Do not give critique where critique is not asked, do not gossip about riders/horses, don't accuse people of "abuse" just because they do something you do not approve of. I have made all of these mistakes as a youngster, and I want to pass on my experiences to new riders. Always use the tack appropriate for the horse. If you are concerned about anything the horse wears (bit, saddle, etc) bring it up to your instructor respectfully. Most of all, have fun. Horse people are very nice as long as you respect their ways of riding, but the minute you get critical or judgy they will judge you back twice as badly. Horse people are HUGE gossips, so stay out of that chain of conversation and stay away of being a topic!
Good luck, and have fun! I hope you stay on HF. I've been to many forums but this is the most welcoming, kindest, and most open-minded bunch of people i've found online. They are so understanding of all styles of riding.