Hi. Last year, I replaced my bathroom walls and ceiling, because of moisture damage to the Greenboard that was installed professionally 6 years earlier. I also replaced the walls in the laundry area because the washer overflowed and flooded the floor destroying the Greenboard walls.
I was concerned about the same things you are--mold, mildew, and mostly the possibility of moisture/water getting through to the walls (drywall). BTW: I did it MYSELF--all of it (even installed a bathroom fan/light/heater in the ceiling, and replaced the cast iron fittings with abs plastic at the washer). No, I'm not an expert or professional--except when it comes to my house.
I won't go thru all the steps I did in removing the tub surround and tiling the tub/shower, because you have a specific question regarding wall choice. So...
In the bathroom: for the walls and under the tile, I used Custom Building Products WONDER BOARD 5'x3' x 1/2" Backer Board, (purchased at Home Depot). WONDER BOARD has a SMOOTH finish. (HARDIEBACKER cement board does not have a smooth finish.)
For the laundry area: for the walls I used WONDER BOARD and DUROCK Brand 5'x3' Cement Underlayment and DUROCK Brand 5'x3' Cement Board [purchased@Lowes]. I used all 3 types because it was what was in the garage and I find there is no difference.
To attach the cement board to the studs, I used a drywall bit and 1 1/4" coarse drywall screws.
For the wall seams and to cover the screws:
in the bathroom, I used thinset;
in the laundry area I used drywall tape and joint compound.
Where the walls meet the ceiling:
in the bathroom and laundry areas, I used 100% silicone paintable caulking--it's a great trick as opposed to using drywall tape and joint compond or paper-faced metal corner bead and joint compound.
After going over the joint compound 3 times (smoothing/sanding with a screen and spreading/fanning), I painted 2 coats of KILZ primer and then 2 coats of 2 coats BEHR interior/exterior high-gloss paint.
The walls and ceiling are smooth, water proof and look great.
Hope this helps. Good luck on your project.
P.S. I have not heard anything bad about Greenboard but, from personal experience, I prefer the WONDER BOARD for moisture/water applications.