I worked H.S. & College summers as a drywaller in NYC back in the 1980's. The last home I renovated, a 5 story double duplex, had an unfinished basement. Framed in metal studs then rail-roaded the sheetrock (4' tall 8' wide) The bottom four feet was a special order non-cellulose gypsum board wrapped in fiberglass - NO paper. Mold needs organic material to grow on. No paper, no mold.
As for your bathroom ceiling vent fan, does the bathroom include an outside wall? If so, vent it our between the overhead joists to the outside of the house with a shielded side vent.
If you find the sound of a screw-gun soothing, and I mean, Who doesn't. Then lowering/leveling, isolating your ceiling with metal furring strip (a.k.a. Chicago bar) mounted perpendicular to the overhead joists is handy. You can feed new wiring over the furring strip for lighting and any other electrical accessories. Need a recess for a light or vent run? knock a hole with your hammer or cut it with a hole saw.
As for any dodgy looking looking studs, Fill a spray bottle with half bleach & water, and saturate them. And again. Then assess once they've dried out.
If you want the bathroom to last, you have to have some kind of membrane in the wet areas. before the technology in a can got to be amazing, I used to spray some 3M 77 glue to the studs, apply a plastic membrane horizontally working from the bottom up so the top overlapped the bottom (think siding), then 1/2' cement board over that, then tile. That way, any moisture that soaked through went down, and eventually back through. Not perfect, but better than drywall. Nowadays, you can put the membrane on top of the wall sheeting which is ideal - whether that's in a can or using a "system" is up to you and your wallet or credit card.
IF you decide to replace drains with PCV, get the thicker wall section stuff or plan on sound insulation. Heavy Cast iron Drain pipes are QUIET. High density is sound abatement. PVC is much lighter and much louder.
Death to all galvanized pipe. It rots from the inside out and then, years later, falls apart if you look at it funny.
If you are replacing the bathroom floor plywood, then use blocking between the joists to tie them together and make them stronger, then using PL2000 or a similar construction adhesive, glue and screw the plywood. If you plan on applying tile, then lay a scratch coat of polymer modified thinset as you screw down some 1/4" Hardi backer on top of it. If you want to put an electric radiant heating grid in the floor, now is the time. Then pour a bucket or two of self-levelling on top and lay your tile. You program it to warm your tootsies in the morning. The self-levelling stuff is expensive, but worth it. just make sure it can't leak through anywhere before it sets, because it will.
Since you have the bathroom apart, make sure the plumbing is properly vented. Gurgling or poor draining or lingering smells are signs that it isn't. The vents are designed to break the vacuum that occurs when a slug of water flows through the pipe. The elbow cast into your toilet base, and the elbows under your sinks are intended to always be full of water to prevent gasses from venting back into your home. A proper vent nearby ensures they don't get vacuumed down while water is draining. The vent breaks the suction.
And if memory serves, bathroom sink minimum drain diameters is 1-1/4", kitchen sink 1-1/2" and tub/shower minimum 2". Feel free to double check, it's been a while.
Another random mistake people made before YouTube: Mount the tub directly to the studs before the wall board is installed. The wallboard comes down just to the top edge of the tub flange, not behind it nor all the way down to the horizontal portion of the tub. The former will allow water behind the tub and the latter encourages capillary action to wick up inside your wall board material.
Have fun!