Thanks LexieCat and hopeful4. I have a meeting set up with my lawyer next week and will have a list of questions about the procedure, what to expect, etc.
It's true that it's just dumb luck nobody's been seriously hurt yet. When I look back on the last year and a half and the pattern of crises, it's become clear that Kid's life has been calmest when she was living with me all the time. Whenever we tried to increase her time spent with her father, in the hope of getting back to 50/50 (hope which I have now abandoned), it was always cut short by bailout phone calls, police at his residence, psychiatric treatment, fights with ex-wife-#2, etc.
Things have been calm since the drunk-driving incident in May (which I'm sure will come up as a reason for why he should have joint custody - because "look, it's all been fine for three months, the longest period without a crisis since January 2016!") - but I would say that's because he's only seen Kid for two hours at a time a couple of times a week since May, so he's had limited opportunities to screw up.