Before tiling he would then roll-on a waterproofing membrane on the concrete board.

If you tile, you can put down a crack isolation membrane down under the thin set. 

The peel and stick membranes are rated for 30+ days of exposure and they have two layers.5) I have had roofing contractor quote me for GAF Timberline, Owens Corning Duration, and Atlas....

It's going to cost 1.5x or 2x what you projected.You will need waterproofing and basically build a "tub" of waterproofing membrane. 

Roof replacement shall include the removal of all existing layers of roof coverings down to the roof deck.Exception: Where the existing roof assembly includes an ice barrier membrane that is adhered to the roof deck, the existing ice barrier membrane shall be permitted to remain in place and covered with an additional layer of ice barrier membrane in accordance with Section 1507.1511.3.1 Roof recover.

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.