The vent can go horizontal below the window, but will need what is considered waste fittings below the flood level.
Going up from the sink, you could start with a med 90, and then when it turns up toward the roof, a long turn 90.
You would add a 2" cleanout below the trap arm.
A kitchen vent could be 1.25" but where would you find the right waste fittings for it?
Everyone else uses 1.5" pipe. You can drill a 2.125" hole through the studs.