If things are as bad as you say the are, I would not take lessons there. Any trainer that allows those things to take place, is not someone I would welcome instruction from.
6 months ago I moved to a barn that allows people to ride during lessons. What I’ve learned from this experience is that I’m much more accustomed now to my surroundings, and when I go to shows the warmup ring is no longer a source of contention for me as I’m used to all the commotion. I don’t think this is a bad thing at all!
In our barn, whomever is in the lesson has the priority. So, if the lesson group has the rail and I need to pass right hand/right hand, so be it. It’s my choice to travel opposite of their direction, and it’s my job to stay out of their way.
When the lesson group is jumping, we either stay on the rail or stay at the opposite end of the ring. There is no interference with jumps or lines.
I love that I can ride with the lesson groups, as we have both a covered and a regular ring, and on rainy days it would be terrible to miss a ride because of a lesson. Likewise, I truly appreciate how prepared it has made me to ride in traffic at shows.
As for left hand/left hand, I grew up learning pony club rules, but not everybody has, so many aren’t familiar with this etiquette. So, instead, I just keep my eyes up and watch where I’m going and navigate in ways that keep my horse out of direct oncoming traffic.
Wait until you get into the warm up ring at a show. You’ll be happy to get back to kids cutting you off instead of the weirdos jumping the jump the wrong way, ponies careening around out of control and riders just chilling in front of a jump with their feet dangling despite being asked to move. That’s not even including the oblivious riders who just can’t seem to see anyone else in the ring and just walk around wherever they want to go and turn in front of you without notice.
When you have a horse that spins on a bad day in heavy/chaotic traffic or if a horse passes too close due to an accident (prior to you getting them), the situation the OP has described sounds like your personal nightmare. That being said, you also learn how to ride in a busy ring to avoid said spinning and staying out of too immediate of a head-on path with others. Circling and a lot of changes in direction is a big part of that.