Is this Sportsplex in Landover, MD?
If it is, I was a parent of a team boy, not girl, but they do very well in competitions overall on both the boys and the girls sides. They have good coaching, I'd say but there can be personality clashes. I'm not as familiar with the personality issues that might be there among the girl coaches... I didn't pay that close attention. I'm not alluding to anything serious or at all like a safe sport violation , but people are different.
Yes that paper you are holding is recreational. You have to tryout specifically for team and then be invited to team. There won't be any skills sheet like that that you'd have your hands on. Their tryouts are usually in the summer and by now new kids have been invited to the team and are already attending team practices.
No, you won't be able to switch between rec to JO team unless you are very young and also very talented. You might be able to switch from rec to Xcel with a little more leniency with the age, but you'll still also have to be pretty talented. Again, i am saying this from what I saw secondhand for your situation since my kid was a boy and not a girl. But you should ASK about this point blank as you will get better information from the Team director than secondhand.
That said, several years ago there was a boy who started in the rec program as a teenager and did move to the boys JO team, as he was extremely talented. So it is possible to be an older child and move from rec to team... but even so I see that as a vanishingly unlikely event on the girls side.
No they don't offer private lessons. There actually is a coach who does private cheer instruction and operates his classes out of the Sportsplex gym. That's as close as you'll get to private lessons there. I actually don't know how to find out about his classes, but if you ask at the help desk or ask someone in the gym they should be able to help you out, or at least tell you what nights during the week you can come by and speak to him.