I only know the USAG rules, I am sure they are different with every league..
A sanctioned meet has a USAG certified meet director, who by applying for the sanction and paying the fee, agrees to run the competition by USAG rules- start/end time, # of competitors per session, # of judges, etc. The coaches and judges on the floor must be a professional USAG member with background check, safety cert, etc. The sanction also guarantees your event secondary insurance.
The sanction is also required to "Count" the minimum AA qual. score per level (in order to score out), as well as qualify to sectionals, state, etc.
I guess an athlete could compete all of those levels at a sanctioned meet, as long as the sessions were in order, competing and getting te required mobility score in Level 5, then Level 6, then L7, etc. This is very, very rare and would be for a very unique sitauation, and Im sure it would have to be cleared w/ the meet director beforehand.
As far as attending a non-sanctined meet in order to try out a level, I'm you possibly could USAG rule wise, but that would be up to the coach/ gym and their policies. I really can't think of any benefit for an athlete to go and "try out" a level- because in most program all the athletes are competing at their appropriate level and would not be as successful in another level.
Sorry so long- hope it helps.