1. Don't leave your hand more than 30 minutes.

2. Don't leave your hand 30 minutes and more.

(Does #1 mean #2? Does 'more than 30 minutes' include '30 minutes'?)

3. Today I got/had a haircut.

4. today I had/got my hair cut.

(Does #3 mean #4? Can we use both 'had' and 'got'?)

5. I won't go to that hairdresser again.

(Does 'that hairdresser' mean 'the person who cuts hair' or 'the hairdresser's shop'?)

1 and 2

I don't understand what "leave your hand" is referring to. A game of poker??!! =)
The phrasing in #2 is not good; go with #1.

3 should be "Today I got a haircut."
4 should be "Today I had my hair cut."
The two verbs are interchangeable, I guess, but I'd use them differently.

5 - Yes, you're right about what "hairdresser" means.