My question is can I use this current horizontal pipe as the vent with the waste drain going straight down?
No, that's not go to work, because the horizontal pipe is too low (it would need to be 6" above the sink flood rim) and because it's pitched the wrong way (your proposed 90 would hold water if it ever go in there).
I suggest you stick with the current layout and just lower the pipe. Cut out the horizontal and the stack low enough to work. You could cut it so that your new san-tee goes directly onto the cut, and then you'd end up with two banded rubber couplings on a straight section above the san-tee, or maybe just one banded rubber coupling if you are able to deflect the free end of the upper vent pipe out of your, i.e. 2" or more.
Then instead of replacing the existing full height studs, just sister them. The cripples only carry the weight of the window itself, so you could skip sistering those, or you can sister the single cripple. The tricky part is that your king stud is very close to the next common stud, so you'll need to sister that with some angled nails or structural screws. Now you can drill a new set of holes for your lowered trap arm.
I was going to suggest reinforcing the studs with Simpson HSS stud shoes, but there's a couple obstacles. One is that the stud bay between the king and the common will be too narrow to access all the fastener locations. The other is that they don't make one for a quadruple 2x, they stop at a triple. You could cut out the lower portion of the cripple against the jack, as it's doing very little, as long as there's no siding attached to it, or you can deal with the fasteners poking through.
Or your other option is to rely on an AAV for the kitchen sink venting, then you can just cut and cap the current horizontal trap arm, and drop the sink drain down through the plate and reconnect the sink drain there. If you do that and there's a 2" or larger drain down there, there's an advantage to making the sink drain in the wall 2", with a 2" cleanout.
Cheers, Wayne