3 feet is a bare minimum. 

Assume the stairs go down to a basement. Have you thought about relocating the stairs? Move the new door opening left (in the sketch) and align the new stairs (turned 90 degrees) with that. OR align the new staircase with the front door. Maybe not easy from a structural standpoint, but the new kitchen/living space would be amazing. Consult a qualified engineer to be sure.

New door location looks a good idea, but you should consider moving it left in the sketch to the corner, as any space left of the door will be wasted. You need to 'control' traffic in and out. A U-shaped kitchen would deter foot traffic coming in the new door from going thru the work area and provide a staging area for things going out to what I assume will be a patio or deck.

Tell us about your family- amount of traffic makes a difference.

Which end of the existing staircase is the upper end? I guess it's to the right in the sketch.

Assume the Master Suite area is to the right in the sketch. Easier to move windows than plumbing, so I would start with the new bath adjacent to the existing one and go from there. If you are going to do some living space outdoors, you may want a door out from the Master- maybe where there's a window now with a Sitting Area just inside?

Remember- the more you try to make the Master Bath all grand & glorious (big shower, whirlpool tub, etc.) the less room you'll find for that closet...

You didn't mention it, but have you thought about opening the wall between Living and Dining? Shouldn't be a load-bearing wall, and sharing that little-used space will open up the Entry, make the Living space feel much bigger, and provide significant expansion to the Dining Room for your once-a-year Big Feed.

A lot depends on structure of the roof. Is it stick-built with joists/rafters or do you have trusses? If trusses, you have much more flexibility with interior walls.

Hope some of this helps...