The downstairs neighbor came up to tell my tenant that he has a leak in his bathroom ceiling, which is right underneath my condo's shower.

Has water come out to scan for leaks? 

I have a unit that has a small leak in the flashing outside the ceiling.

But understand the type of risk could be the exiting tenant having poured cement down the drains, no appliances, no cabinets, a leaking roofs with mold inside the house, no AC,  or squatters having lived in the house with untold 'grossness' being left behind for you.  

There's a lot of great advice here, but the first thing I noticed was the dark photos that look like they were taken with an iPhone.

You do not want to be 4 weeks behind on a roof leak or mold issue.

it sold for 2.2M.the property was a huge project also... leaking roof, mold, plumbing, lots of termites, 1970s everything. 

Not to mention I ditched my iPhone 5S for the Samsung Galaxy Note 4 (battery life = sooooo much better, lasted TWO DAYS without charging) and I don't have iTunes on it (and I probably won't install it either). 

What type of work is typically recommended for an older (mid 90's) single wide mobile home roof that is not leaking but that is very old looking? 

We had a surprise at the end when we were getting ready to re-rent and had gas leaks in all of the units.