Depends on the club - but probably most do require that if you're on the team that you compete. The reason that they say is that because you are paying less per hour of instruction (even though really you do spend time warm-up and conditioning that isn't so heavily coached at all the gyms) and they want something back in return - competing for their gym, make their gym look good, etc. Sounds kinda bad in a way...I know.
For results, gyms may set a certain score you have to get during the season, and maybe you have to get it like 3 times or something, to pass to the next level and compete for the next level. The state does set a score as well, but your club's might be higher. Some gyms like to keep their gymnasts in a level maybe 2 or 3 years for unreasonable reasons...sometimes gymnasts are better at optionals (levels 7,8,9,10,elite) than in compulsories (levels 4/5/6) where everyone has to do the same skills/routines that are set, but they keep the gymnasts in the compulsories perhaps because they didn't reach that score they set, or they did reach it but haven't learned the next levels routines totally (they don't want they to not compete for their gym while they are learning the new skills), or they think they will get a better score for the gym at level 9 rather than level 10 for example, etc....
Smaller gyms or gyms run by a certain kind of management (educated) will be more reasonable and flexible.