Adequate information - Do you think principals have adequate information?

About the relative quality of teachers? Yes. They absolutely do as the monitoring/mentoring of teachers is a prime function of a school's administration.

If so, why couldn't they just provide it to the parents then?

For the same reason that any employer doesn't publicly release performance reviews to their clients? Are performance reviews public information in your work place?

Besides, there are many tools such as feedback reviews + word of mouth which would help with the information asymmetry.

Ratemyteacher/Ratemyprofessor have been around forever. They do

Unbalanced classes - Classes may currently have balance within the school, but there is already a massive inbalance between schools.

Your comments were not about balancing students/teachers between different schools so I'm sure what you're going for here.

And why do we assume that all classrooms should have an equal distribution of IQ? Colleges do the opposite - they purposely exclude lower IQs.

Higher education and k12 have intrinsically different missions. Public k12 schools are legally required to teach every student who walks through the door. They are also increasingly punished for failing to show educational growth for every student, regardless of that student's abilities and/or desire to learn.

Colleges not only get to exclude students that don't meet their requirements, they have no obligation to see that those students learn. If a student fails a college course it is patently assumed that the failure is the student's fault. K12 education isn't provided with that luxury (appropriately, IMO).

Difficult students going to bad teachers - Maybe this is how it would turn out. I'm not sure.

If you're creating a competitive, zero-sum game. Your wealthy/highly engaged parents will already assure that their students will be well supported (and take active benefit of legal devices to ensure that they are e.g. medical referrals/504 plans/etc.). By creating an explicit mechanism to further sort students based on their ability to bid financially for particular teachers you are intentionally skewing the playing field. You may not be sure, but I have zero doubt that your scheme - intended to "punish bad teachers" - would result in difficult/poor/reluctant students being placed into one class with a teacher who is inexperienced or otherwise less popular. Were that to happen, it could very quickly become a violation of the Equal Education Opportunities Act. Good luck at that rodeo.

(But even if it did, it would be even more of an incentive for teachers to perform better - just a thought).
I would be hesitant to predict any specific outcome of this scheme, only that teachers would be more inclined to provide the service that parents actually want.

Dude - five tough students in a class of thirty is already a huge hurdle, even for experienced and professional teachers. Your system would not incentive teachers to improve. They would simply quit.

You're also assuming that there is some large deficit of effort between successful and unsuccessful teachers - as if unsuccessful teacher are just kicking their heels up on the desk all day. This is fallacious assumption. You may not believe it - but teaching is a complicated, difficult job.

Ethical grounds - I would question the ethical resolve of any school that isn't open to whatever system will help students best.

If the system itself is unethical then no ethical professionals will employ it.