accident / mistake sound like undesirable consequences, luck / chance sounds like a desirable consequence.

They are slightly different. "Chance" used here would mean that it was a random event, but does not indicate how likely, or not, the event was to occur. "Accident" is more like something that should not have happened, but it did anyway. This indicates the event was unlikely to happen or at least unexpected. "Mistake" is similar to accident, but it implies that something went wrong to cause the event to occur. Perhaps you missed your train, and that caused you to meet your friend at the train station.

I think, they are not. "By chance" and "by accident" means randomly, without intent but action "by accident" usually causes some undesirable consequences. By mistake - as a result of some not intentional but wrong actions