It makes me nervous to go ahead with it, but moisture checks in the unit downstairs following heavy rains since the time of the water intrusion (including a check 2 weeks ago) have been fine.

If it’s unfinished then it shouldn’t be a problem but if you’ve got a sub floor and walls that moisture will get behind and underneath them there could be a concern but also if the price is right it doesn’t take that much to rip it out completely.

i will say that in my experience, water /moisture issues cause majority of damages. 

The report says that it has polyethylene plumbing and organic growth on some spots on the kitchen ceiling, but they don't show elevated levels of moisture.

The landlord would be responsible for any other lights on the property that benefit both tenants.If is your building, your moisture problem and if you wants to run a dehumidifier 24/7  you will have to pay the electric bill and the cost to run 24/7 is about $300 to $400 per year depending on the size of the dehumidifier.If you want to run a pieces of equipment that expensive then you probably should install an owner's electric meter and that will probably cost $4,000 to $5,000 with the additional wiring you will need.

You still got issues with maintenance of an older home eating in to your cash flow as in any older home (and cash flow might be your only benefit)look at the age of the 1) water heater2) AC3) appliancesGet an inspection to see regular rain penetrations, failing mechanical and plumbing, so many things , its worth the money.

It has a rather dank smell but no visible moisture or mold.

It may take some phone tag to find the right department to ask the question and you may have to try from a few different angles to penetrate into the REO department.

He said there was some moisture under the house that probably caused it. 

My upstairs units are now collecting moisture through the attic.