I am 99.99% sure that is not a load bearing wall and the header is unnecessary. Who framed this? You cant bury a splice inside a wall so you need to go back to the closest box and run a new longer cable over the door frame.

Im not planning on splicing it inside the wall. Was going to run a longer cable from bluebox. How can u tell its not load bearing?

Look at the roofline. If the wall goes across the ceiling joists, it is most likely load bearing. If it is parallel with the ceiling joists, it is not loadbearing. Most stairways are set parallel to the ceiling joists and box framed. The only thing you are supporting is the ceiling joist and some drywall. If you put a stairway across a load bearing wall, you lose ceiling and roof support because you will cut 6-8 or more ceiling joists and have to use a much more complicated support system.

What the picture doesn't show is that there is another stairway that goes 90 degree with the picture. So multipke stairways each going perpendicular to each other. The ceiling joists are perpendicular to the wall in the pic

The width of the header is determined by the length and type of load it has to carry.
It's very, very unlikely that a header of that mass was needed in your situation. Have the builder look into that. Perhaps he can replace it with one more appropriate which will allow the use of cripples. That will make your re-wiring much easier.

Looks like you need to repair drywall anyway. I would relocate Boxes away from door frame. With power off, identify both sides of both wires(colored tape, writing on wire, anything to keep track). Cut wires in middle of door frame, then route 2 cable from box to box(use large boxes for room. Reconnect.
Don't notch. I thought code was 1 1/4"? From edge of stud, if less, you can use nail plates. Can you drill down to underneath and back up?