In the future, just tell the tenant to limit the use of the kitchen and put a bucket/bowl under the sink where the leak is. Don't cut off someone's entire water access.
Don't take the tenant's word for it, when you get home, make sure you check it out and confirm it's repaired to your desire. I wouldn't downplay this leak. It could have been going on for years and may be further damage. For sink water waste to get to your floor, it must get around...kitchen cabinet base, 2nd floor flooring, subfloor, floor joists, insulation, then your ceiling tile). This was no small leak. Make sure all of those items aren't damaged.
While the ceiling tile is off, examine the space between the 1st and 2nd floor. Look for any rodent waste, wood rot etc.
What is wrong with the doors? Doors are very tricky to replace in older homes and generally very expensive (especially exterior doors). A little bit of sanding and paint can make them look brand new. Perhaps a picture could help me understand your concerns. This would be much lower on my to do list.
So what is the living situation? Is the tenant who didn't pay now in the 2bedroom and you are taking over the 3 bedroom? 1 bedroom is still vacant?
At the time that I noticed the leak I had no idea where it was coming from, and I was on my way into work for a 12 hour shift. The water has been off to both of those apartments for the past several months; I only just got it turned back on Friday. The tenant understands that he's living in an apartment that is in many ways a project in progress. In truth he's only half living out of the unit for the time being; he's been staying with parents most nights due to aforementioned lack of water. And he was outside working in the garage when I found the leak, so it wasn't as if I just turned his water off out of the blue; I went over and told him "hey, there's water leaking from your apartment into the downstairs bathroom, I disconnected the water until we can figure out where it's coming from". He has access to both apartments, so he was able to turn the water back on himself after fixing the leak.
As for the damage, I haven't seen the upstairs unit yet (though replacing that kitchen flooring was already on the to do list, so further damage to it is largely irrelevant), and there is no insulation between the two floors, at least not in that area. As for the joists, they were definitely wet but did not appear rotted. As I said, the water has been turned off to both of those apartments for several months, so nothing has been leaking anywhere recently, not until I turned the water back on Friday, which is likely when it started.
As for the exterior doors, they're mostly just in bad shape. They don't close easily, and the portion of the frame where the strike plates are mounted is really chewed up on most of them, to the point that I didn't even bother replacing the strike plates because I wasn't confident I'd be able to get the new ones to screw into the existing holes. I used a 4 foot level to see how level the frames are, and they're all surprisingly close to even. I *think* that with a bit of shimming pre-hung doors should work. I'll take some pictures and post them, would love to get some more experienced opinions on my options.
The living situation at present is:
1 bedroom unit - Vacant
2 bedroom unit - Occupied by the tenant formerly known as "deadbeat" (who turns out to actually be a pretty swell guy, time will tell if he pays his rent when he has a good landlord)
3 bedroom unit - Vacant, soon to be occupied by yours truly, yours truly's SO, and roommate who is renting 1 room for $300/mo
1 car garage - Being rented by tenant for an additional $200/mo
