Some guys do a basic magnetometer scan (detects metal) and others can do more comprehensive ground penetrating radar (GPR).  

Seal all roof penetrations.

By that I mean anything from moisture testers to infrared photography for heat loss.Well worth the money.John Rogers

Otherwise, significant moisture will be residing around your basement walls for prolonged periods. 

Buy your own infrared camera to inspect the internal moisture levels? 

Typically when there is a perfect line cut around the drywall it is a sign of some type of moisture issues.   

Personally, I would put the tub on the lower wall, and the toilet on the same wall as the sink, to the left of it (leaving the sink where it presently is).The placement of the window within the shower would seem to create far too great a potential for water penetrating the wall around the window, not to mention that it just seems odd (perhaps that is just personal taste)

I would think laminate would be your best option, last longer than carpet and less maintenance.  1) Depending on the laminate you use you will need a moisture barrier, the people at home depot should be able to help you.  2) Yes any decent contractor/handyman should be able to knock it out in a week if thats all they have to do there.3) Not familiar with your area but you could expect to pay anywhere from $1.50-$3.00 per sg/ft.  

I couldn't really figure out why, maybe covering some past moisture. 

I'm currently binge listening to the podcasts and look forward to meeting other members, specifically here in SoCal, which is considered a notoriously expensive & tough market to penetrate.