To eliminate the problems, a contractor recommends that M put a new rubber membrane on the worn membrane.

They have a membrane called kerdi that looks like u can put over any type of board.

I've never removed a water softener from outside though and it seems a mower could easily clip a loop if the weeds/grass get too high (based on your pic).In Las Vegas I've also experienced membranes of older water softeners breaking which pushed tiny yellow resin beads through the plumbing causing interior plumbing issues.

We ripped off two sections of EPDM on a French Mansard and went back with a tar/membrane roof, adding two inch insulation panels on a 2500sf house in 2005.

These roofs usually are tar with gravel, or some type of membrane system.

They put the membrane outside the foundation and had the piping and crushed rock buried outside of the house.

With respect to the rehab it really needs 2K as new flooring(uneven) New linoleum in bath/kitchen, new bath fitter, minor bathroom plumbing, new paint, two new windows, new dry wall and a rubber membrane on the upstairs porch are required.

However, they did not sit DIRECTLY on the roof, instead they had a heavy, rubberized mat that was placed on the roof first, to protect the actual roof membrane.

Flat Membrane and rock versus metal with slow pitch.    

I consider myself a savvy apartment investor and I have zero problems with flat roofs as long as it's a newish TPO or rubber membrane roof and installed properly.