Alice dance is not a quest, is a minigame, which yes, you can continue to trigger as is the way to have Alice show up at the bathroom. Might change once her quest continues in later updates.
What part? As is know this quest has oddities and it just doesn't really end as her quest is not completed.
Is not that it cancels the cooldown, is due to the way the variables are saved, for now this is not to be abused as it can cause issues but yes, outside of things like the flower shop which was able to be correctly saved, most of the variables triggers are reset for the day, same reason you can take pics again at the kitchen or go to the church again on Fridays.
The minigames issues are known and will be fixed. Same with some overlap expressions and Liz at the bar, which doesnt just happens with Liz but other events have a slight chance of trigger this effect as well.