However, about 10 feet away, not near the laundry, I also noticed some moisture underneath some of the vinyl flooring.

If mold is on the joist that would be no problem to clean, just needs some equipment and on the sub floor i believe pending on how severe it is you may just get away with a cleaning with specialized chemical thats stronger than bleach (I know the name but cant remember how its spelled) , as well as a negative air machine, some duct tape, plastic sheething, tyvek suits, gloves and other ppe needed for the workers or it could be severe enough that the floors would have to come up (I know you dont like the sounds of that) and possibly trace it up the walls if any moisture wicked its way up sheetrock and the insulation thats in the walls?...

If the shower enclosure is in good shape and the remaining tile is just cosmetic pull it off, repair any drywall damage, and cover it with a shiplap or an accent wall panel (just make sure you use moisture-resistant materials). 

There place is musty and dank and probably some terrible moisture and water damage in the walls.

(to include moving/removing the lath boards, new drip edge, valley metal, sheathing (w/radiant barrier), re-doing the penetrations through the roof (if gas) etc etc etc)To Greg: Whatever the case, see what the extra charge is to use a dimensional shingle instead of 3-tab.

Chlorine constantly escapes through the plastic walls of its containers. (4) The ionic structure of bleach prevents Chlorine from penetrating into porous materials such as drywall and wood---it just stays on the outside surface, whereas mold has enzyme roots growing inside the porous construction materials---however, the water content penetrates and actually FEEDS the mold---this is why a few days later you will notice darker, more concentrated mold growing (faster) on the bleached area

Forced air is the way to go to prevent stagnant air build up and to prevent mold/moisture problems.

It seems like I'll need to do at least two scraping sessions to remove it...one to take off most of the paint so the water can actually penetrate and soak in, and then a second scraping to get the rest of the texture off.

We had a mold inspection done and there seems to be mold and high moisture content in 2 bedrooms, the kitchen and bathroom areas.

9) Flooring10) Trim and detail work11) PaintMOST IMPORTANT - 1) Egress/Ingress - you must have the proper sized windows in the bedrooms, and an entry door that is to code 2) Moisture - since you're in a wet rainy area, make sure you have this covered firs thing!