Beam too low in bedroom?

[Leland S.]

Hi I'm doing a rehab and pulled out walls of a load bearing hallway to combine two bedrooms to a master suite. Problem is a few things, the floor upstairs feels like a trampoline because the old owner who had built the upstairs addition used 4" floor joists instead of 6". My contractor says he can double them up to make them more stable which I hope is correct. 

The other issue is I almost hit my head on this new support beam they put in the ceiling where the hallway walls used to be, but they added it to two 2x4 header beams rather than removing all of those 2x4's and replacing with the beam for some reason. So now the beam is about 2" below the doorway entrance making an odd entry beam lower than the entry door. I think this is unacceptable. My GC says he can cut away half of the beam if we add a closet to the corner of the room which we planned to do up to the beam. 

Any second opinions would be helpful.