You can begin a girl's questline if you have the requisite number of stars AND have finished the previous girl's first quest.

The ticket event lasts for one hour after you finish a girl's second quest, allowing you to easily farm third episode battle tickets for that hour. After that, you have to farm battle tickets like you normally would: spend 20 energy in her first two quests for a chance at one battle ticket.

Unfortunately, it seems like getting battle tickets as a reward is bugged, so getting one is basically a waste of energy or tickets.

All of this is personal observation; nothing official.