So some answers to some questions and an update:
-there is no lease. The landlord said she was going to have them sign one, but she never did. This is not a professional landlord situation. it's a friend of a friend renting out a house. Obviously, it's not ideal.
-using the A/C causes further damage, which is why the LL requested that it be turned off. The water is rising and extending. It has damaged some of my friend's possessions at this point (minor stuff, but still a big bummer.)
-the LL only uses friends to fix stuff. I'm assuming as a cost-saving thing. That's why it has to be this one specific company. My friend requested that she either send out a different company who could come right away or be allowed to leave the air on until the repair company does come.
So the update is that the LL relented to letting my friend continue to run the a/c even though it's flooding the house. So all told from calling to let her know it was leaking, to the repair people's EXPECTED arrival (they aren't there yet) it's about 48 hours. it's better than the 4 days the LL originally requested.
My friend is looking for another place ASAP. Her big concern is that the LL will keep the deposit out of spite and it will be hard for them to forfeit that much money. It's also really daunting to try a last minute move when you're 6 weeks away from your induction with 3 and 1 year old children already at home.