Is it permissible in congregational prayer for a worshipper to lean on someone else and rush through the prayer when they have nothing to do afterward but play?

This person annoying you by leaning on you during prayer is impermissible if it causes you harm, and he should be advised gently. As for his rushing in prayer, if it compromises ṭuma'nīnah (tranquility/composure), then his prayer is invalid. But if ṭuma'nīnah is achieved, then his prayer is valid. He should be advised gently if the reason for rushing is merely playing around.