Is it permissible to play video games and tennis and hold competitions inside the mosque, and is reviving the practice of night prayer every week and a night called "Katibah" every month (eight rak'ahs of congregational night prayer) considered an innovation (bid'ah), and is it permissible to pray behind those who do that?

Playing in the mosque is not recommended. Designating a night for Qiyam (night vigil prayer), if it is due to a belief in its special virtue, is disliked (makrooh). If it is not due to such a belief, then there is no harm in it. As for dedicating a specific night each month to a particular prayer, there is no basis for it, and it is feared that it might be an additional innovation (bid'ah). You should advise them and forbid them from doing so. Praying behind them is permissible.