Is it possible to have figure skating lessons (private) independently on open skate, or will I get in trouble

You can definitely do this on Mon, Wed, and Fri from 12-1:30pm. Other times, it's more crowded, but it shouldn't be a problem.

They do have private lessons. I think they have the instructors. When you contact them, they assign the instructor.

I would call and ask. I know for tennis no outside coaches are allowed, unless they are family members