Do I need to remove existing drywall before framing an interior wall?
I know you already accepted, but I disagree with @bib on a few points that don't fit in a comment, so I'll provide my own answer. 


So long as you build a solid frame and connect it well, once installed there is really no chance of existing drywall being "crushed" by the frame. To crush the connecting drywall after installation, you'd need a very serious lateral force which is enough to flex or shift the frame itself. Since your wall connects at both ends, it would pretty much be impossible to put this lateral force on the new wall.

You of course have to be careful during construction since it is possible to crush what's there now. 

Wall construction

Solid construction means using 16d (3 1/2") nails (I prefer spiral to common, or use a pneumatic framing nailer). I'd also connect it into the existing structure using 3 1/2" #8 or #10 screws (not nails). Screws are stronger, and trying to hammer in nails is going to cause that lateral force that may damage the drywall (as @bib suggests). 

This includes screwing into the subfloor and ceiling. The subfloor should be plywood, so to prevent it from shifting you can attach anywhere with 2 1/2" screws. The ceiling will just be drywall covering joists; if the joists run perpendicular you should be able to anchor to each one; if they run parallel then if you're lucky there will be one right over top (likely since it's at a corner) and if you're unlucky you'll have to add framing. 

There is definitely a stud on the top end of your new wall (since it's a corner) so just be sure to screw into that. 

3/8" vs 1/2" drywall

There are, however, a couple things I'd point out regarding the 3/8" drywall you mentioned: 


Standard 16" O.C. framing uses 1/2" drywall. If you put 3/8" on 16" O.C. framing, it is not going to be strong enough and will flex if someone leans against it. I personally would only use 3/8" for boxing in a soffit on the ceiling of a basement or similar. 
If your existing drywall is 3/8", then I'd be worried if the new corner joints (mostly the one at the top right in your diagram) will be strong enough. Any flexing on the walls in the corner will cause cracks down the corner. You may need to put some framing in behind to support the drywall - this means opening the wall there.
Even if your existing drywall is 3/8", I'd still use 1/2" for this new wall. 


Speaking of drywall, I'd remove the corner bead on the existing outside corner (in your drawing: at the top left of the new wall). This will make it easier to tape and finish the new drywall; you can treat it like a butt joint. 

To open or not open drywall

On the bottom of your diagram, if there is no stud, then you have two choices really, and to me it depends on a few things. 

If it's an interior wall, it's probably easy enough to just open the wall and add a stud or horizontal framing between existing studs. You can do this on an exterior wall as well, but you additionally have to deal with insulation and vapour barrier. 

Attaching a wall to drywall without a stud

If, and only if, you have a solid attachment point in the floor and ceiling (meaning joists) within about 1' of the wall edge, you can avoid opening the drywall. 

Use a few zinc drywall anchors (probably one every 16"). To install this:


Place a single 2x4" on the wall, and pre-drill through it right into the drywall for every anchor point (every 16")
Remove the 2x4" and put the anchors into the drywall
Screw the 2x4" into the new anchors 
Build the new wall same as you normally would, so when finished you'll end up with two studs on this end, and solidly attach the studs together solidly using 2 1/2" #8 or #10 screws (you don't want to apply hammer force onto the drywall). 
Be sure to put a couple 3 1/2" #10 screws into the joists above and below near the ends.


By building the wall this way, you are essentially using the new wall to support the existing drywall, rather than using the drywall to support the wall, which is why the connections to the floor and ceiling joists are so important.

You could do the same for the ceiling in case there are no joists above, however, I would not do both, and I would not do the ceiling this way if the new wall is longer than the length of a single 2x4".