Stud wall bracing

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.

Look where the joists meet the ridge beam where you're going to put the door.

Get some dimensional lumber about 4 feet long, at least as thick as the existing beam. Nailing a couple of 2x6's together would probably suffice in the short term.

Give yourself enough room to work, and using a jack post and this 4 foot beam segment, support the joists in the area where you are going to work. Don't over support, as you'll lift the floor above. The idea is that you want to take the weight off the wall where you're working. Jack the post until it's snug, and take out one wall stud in the middle of the door. Check the supported joists for level, and jack up if necessary.

If required, remove another stud, and adjust the jack again.

Put in your door frame and header, and slowly jack the temporary beam down.