Once the root of the problem is solved, spray the affected areas with a 50/50 solution of bleach and water, and then replace the dry wall with new material.

@Justin Fox - you need to exercise caution with spraying {closed cell} foam directly on the underside of the roof deck as you can both prevent the roof deck from drying (insufficient ventilation) and cause condensation within the roof assembly itself ... leading to premature failure of the roof assembly.

I would have it sprayed  for bugs no matter what you see before humans live in it .

What I tend to do is spray multiple very light coats of shellac over the stain over the course of a day.

If it were mine I (MIGHT) spray it white, and put shutters on it and add windows in place of the boarded up openings.As well as add a see through commercial style door on both ends of the hallway (rape and other attack prevention) if the area called for it.Cement a few park style BBQ's into the ground on each side (split them up to prevent tenants arguing) and park style trash cans chained to the ground (with lawn personnel responsible for changing out the bags).A repave and paint of the lot (might be a lender requirement).A commercial awning with WELCOME on it (in a bad neighborhood not everyplace seems welcome).The building name and address (easy for Police and visitors to read from the road).Some leasing signs and good look at the roof regarding replacement (for leaks) because the building looks neglected.Good bright lighting (up high on the building) around the building and parking lot at night, and no bushes (hiding places for attackers).I said (MIGHT) above, because I'd need to take a drive around the area and see the competition, to decide what I wanted to offer to make the building competitive without overspending.Also, I'd do that drive and crunch the numbers on it before being locked into a purchase contract for it, because it might be a pass instead of a buy. 

In her first month, she mentioned the roaches where a problem but said she was going to do things to remedy it, i.e. moth balls, sprays, etc.

Also, consider spray foaming the third floor attic Or roof line.

@Boots BonnerAttach a hand-held sprayer where the shower head is currently and spray systematically from low to high on the control wall for a few minutes in each location.

There were gaps in the front and back doors big enough to fit a hand under, a mosquito infested back yard and other bugs also which tracked into the house under said door gaps including wasp tarantulas and scorpions... we had a professional company come out 1x per week to spray for mosquitos so our kids and dogs could enjoy the outdoors and lay granulated pesticides (non toxic).

Get a hand pump yard sprayer, fill it with straight white vinegar and spray every bit of the carpet.