You started this with doors trying! things
As for уже, sure it does but not in a sense you might be thinking of. Уже makes the action completed. Yep, 2 imperfective verbs conveying a completed action, albeit a prolonged action that's finally come to an end. If you omit уже from the sentence:
Двери не пытались закрываться.
That'd be a prolonged action with unclear state of completion, like the doors didn't try to close for some time.