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

What you're doing here is called &quot;sistering&quot;. I personally would use 2 screws each at the top and bottom.  More is unnecessary - after all look at the other studs - nothing holds them in the middle.

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?

For the left stud you'll need to toenail - that is - install the screws diagonally into the top and bottom plates.  It'll be easier if you pre-drill the holes - install at a 45 deg angle, 2 top 2 bottom again.

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.

2x4's are actually 1.5 by 3.5.  So you'll be sistering into a 3&quot; depth, so use 2.5&quot; screws for the sistering, and on the diagonal, use 3.5&quot; screws to cover the length of the hypotenuse and give you good bite w/o over sinking.