Welcome to the forums! If you remove the jack studs, you will be left with bubble gum to hold up your header. Short answer, no. I have seen 1x4's replacing 2x4's for interior door headers, but it is really iffy for support. Ideally you will want to remove one jack stud and king stud, the header and sheetrock surrounding it. Move the king and jack to the required opening and replace the header, resting it on the jacks. Stand by, XSleeper will be along shortly.

I was going to say Z or drip flashing above the trim...but since that looks like poured concrete walls...that wouldn't be much help. I agree with working on the deck to try to deflect as much as possible, but a quality caulk should not wear out from water. Maybe cutting a taper on the top of the brickmold trim? Or better yet...cut a beveled piece of wood something like this ( sorry it's so small) and extend it about 2 on each side of the top brickmold. If the brickmold is caulked and the beveled piece on top is well caulked, it would direct any run down water off and away from the joint. Sort of like a window sill only on top of the door.

Thanks for the quick responses, additional fact: Above this door is the deck and the main floor back door entrance. Is there any flashing that would look good at the entryway of the door above (it is white brick) that will deflect the water away? Thanks Wil