Let’s just start this out with one basic understanding - IHSA is a world of it’s own. If you take a lesson with a non-IHSA trainer, not everything will be applicable to IHSA.
Like supershorty said, don’t worry about getting the horse in a frame. If it easily rounds up, you can let them but it’s not worth the fight. Since you are new to hunt seat you probably aren’t ready to feel when a horse is going to easily go on the bit (unless you’re coming from dressage - you don’t say what your previous discipline is ) so just keep a contact that allows you to steer and stop and enough leg to keep going, and you’ll be fine. Sometimes in IHSA this means absolutely no contact with the horse’s head in the air. This is perfectly acceptable in IHSA if the other option is a fight with the horse. (Do not expect this to win in any non-IHSA setting.) Just concentrate on your own position, you might want to pick up a book that can give you pictures of correct form. George Morris, Anne Kursinski, and Anna White-Mullin all have very good books.
As for getting a feel for a horse before jumping, you probably won’t have to worry about that for a little. It sounds like you’ll be put into W/T or W/T/C to start, and by the time you point up into Novice Fences hopefully your coach has had you on plenty of different horses. Most of the time there will be a little blurb about the horse, enough so you can locate them and get a basic idea of how to ride them (“Dobbin - 16.0h chestnut with blaze, simple lead changes, stick optional, keep your leg on, sticky right lead”). You can watch the horses school, as you said, but you won’t know which horse is your draw at that point. Sometimes after you draw you will have time to watch the horse go with someone else, which is nice. It’s even nicer when that someone else is your teammate and you can ask whatever you want ;). Many of the horses are IHSA veterans that upperclassmen may be familiar with, so it’s always worth asking your team anyway.
If you’re on a horse completely cold, just gather what information you can and wing it. If you were given spurs, the horse is going to be lazy - you don’t get spurs for a slightly slow horse in IHSA. If he looks part draft (or if he’s definitely all draft, you never know with IHSA!), he probably doesn’t have a great mouth. If he’s in a pelham, you’re probably going to need it. At some shows you have enough room to walk around outside before you go in the ring, but many horses change drastically once you get through the in-gate.
For a jumping round I like to do this - walk through the gate, note if he changes at all (head goes up, starts pulling or sucking back, etc). Pick up a trot. If you think it’s going to be a fight to get back to a walk, just canter and leave the fight for the end. (You are not training the horse, you are covering up the training flaws, BIG difference and very tough to adjust too). Otherwise, come back to a walk and make a note of how responsive they are to your hand. Wait until a turn to canter, note how responsive they are to your leg. Then I usually let my instinct kick in. If they were slow to canter, I probably won’t choose a long one for the first fence. Have a hard mouth and I won’t take an inside turn if I need a simple change, etc. That will all come with practice, and not something you need to worry about yet
IHSA is fun, but you need to know how to laugh at yourself! Good luck!