Those "studs" are called furring strips. Its not an ideal way to finish a basement because it limits electrical placement and also makes installing drywall a little queer.
It sounds like the furring strips are in place but no drywall.
The furring strips are 3/4" think, drywall is half inch thick, thats 1 1/4" total. They make 1" and 7/8" screws those, or nails, are fine.
I would also consider insulating between the strips. There is a foam board made for this purpose and is 3/4" thick.
Glue under the drywall is a good idea. Greenboard in the bath is a good idea in the bath except for around the tub if you plan to tile. Tile goes over a cement board. Do not tile over greenboard in wet areas.