What are the steps to raising an interior wall header?
First things first. That header is there for a reason. Simply cutting it out could result in crunchy sounds at best, and significant sag at worst (which is difficult to reverse). I'd consider a temporary wall along it to carry the joists above. Use towels as padding to prevent damage. Cut the studs a bit long and drive them in to create lift.

What is the first thing I should be cutting/removing to release header?

Since you've eased the load on the header, it doesn't much matter. If you're discarding the header, cut through it to allow it to drop out, and start cutting nails where you can access them. They're likely along the top of the header and at the ends. 

What tool should I use to make sure all cripple studs are cut in level and straight?

Use a carpenter's square to mark them. Mark the outer ones first, then use a straightedge between to mark the inner ones. For best accuracy, mark at least two faces. 

Use a circular saw to cut them, or a reciprocating or hand saw. Be sure to keep the saw table tight against the stud so your cut is square. They'll be flopping around, so you might lay a horizontal board across them all and run screws in to stabilize. You could also just pull them out, cut new ones, and toenail them to the top plate. 3" gold screws instead of nails can make this type of work less frustrating. 

After I cut cripple studs and remove existing jack studs, what should I do next? 


With a helper (or a temporary support leg), hold the header tight against the pin studs. Toenail it into the king studs from the face and/or below.
Cut your new jack studs to fit snugly. Set the bottom in position and drive the top into position, using a block of wood as protection against damage from the hammer. You want a snug fit, but if you're really wailing on it you might shave 1/16" and try again. 
Nail the jack studs to the king studs. 
Toenail the bottoms of the pins to the header. 


When should i cut my sole plate?

Doesn't much matter. Mark 1-1/2" from the king stud with a scrap block and cut it, or cut it flush later. 

Where can I get studs to match the legacy size of the existing ones?

I'm surprised that a stud from the 90s is undersized (it may have just shrunk a bit), but you can just run them through a table saw, pass a hand planer over them, or ignore the issue altogether. 1/16" on each face isn't much of a problem.