Walls – Wall stud spacing too tight for replacement medicine cabinet

The cripples are typically a continuation of the 16" stud intervals. They make hanging drywall easier since an 8' or 4' length will fall in the middle of a stud on each end (with the exception of the cut end when your room isn't a perfect 4' increment). Like Shirlock says, they aren't required for non-load bearing walls (hardly anything is since the wall isn't load bearing), but as you remove parts like this, other details like the drywall and installing trim will become more difficult.

You need to locate and identify the underlying structure, where and what. Only then can one determine the best attachment method and wall layout. Stairways typically have stringers under them running the length of the stairway run that you can connect to. But there are endless variations on stair construction, so only someone on site can determine this. There are various devices for locating underlying wood structure, their effectiveness varies by local conditions.

Stud framing is typically 40.5 cm spacing on center, so you may need two intermediate studs in addition to the end studs. You need solid blocking flat to the mounting face in order to mount plumbing fixtures. Some hardware needs to be framed into the wall, so you need to identify exactly what hardware is used in order to provide and locate any needed structure.

Unless you can provide all this information, no one can adequately tell what to do in enough detail to ensure proper construction.