What's the proper method for installing new studs in existing frame?

FYI – This is similar to a question I asked last week, but is about something more specific. Expect to see this image yet again in another question I will be asking later today :).

I need to install Genie Clips on the far left and right side of the wall, but there are no studs exposed. The right side of the wall should be easy since there is a stud right along the edge on the inside. All I need to do is put a new stud up against it and screw it in.

The left side is a bit different since the next stud is about six inches in. I am going to need to attach the stud to the top and bottom of the frame.

My questions:

• Right stud: What’s the best way to screw the stud into the frame at the top and bottom so that it’s sturdy enough to support the new wall I am going to hang on it?
• Left stud: Since I will be able to screw this stud into another stud, how many screws should I use? Should I put the screws in pairs so that it’s two screws side by side all the way down?
• What length and type of screws should I use for both studs? I am assuming wood screws for this job, but I am not sure about the small details.
• Edit: Perhaps there is a sort of metal bracket that I would need to install at the tops and bottoms instead of putting the top and bottom screws in at an angle?

Here’s my wall: