the BIGGEST mistake is not getting an instructor.

Since you seem to have that one covered, I would say the next one (BTW, all of the above mentioned are good and should be heeded) is that people tend to think that the plane has a brain.

Example: Plane is flying straight and level. Rookie inputs aileron to start a turn, plane banks, and noses down slightly. They wonder "Why did it nose down? I didn't give it down, I only gave aileron".

You explain why it happens and that they need to give UP elevator to compensate for it. So they give some UP elevator, but they don't realize that the more (or less) elevator you give, the more (or less) the nose will rise.

They tend to think 2-dimensionally - "I banked and gave up so why didn't it make a good turn?"

I like to refer them to driving a car. I'll say, "If you're driving down the street, and you need to make a left turn, how much do you turn the wheel?"

You can't answer that question with a simple answer because every turn is different and your speed and road conditions are a factor. The only proper answer is: "You turn the wheel enough to make that particular turn. No more, no less".

It's the same with turning a plane. Every turn is different. Some will require more (or less) bank, some will require more (or less) elevator, but each one needs to be controlled by YOU.

The next thing is: Do you HAVE to maintain the same altitude during a turn?

No. It's a good practice, but only for honing your skills. Let's face it, once you learn to fly you may want to make your passes over the runway LOW and fast (I know that's what I like to do) so you'll need to lose altitude on the downwind turns. But the same rule applies: How much altitude you lose depends on how YOU performed the turn.