I'd clean the cab side, finish it as you did the other cabinets...
Agreed, that cabinet side needs to be refinished.
Base shoe---not quarter round.
Yes a base shoe is much better, quarter round looks very dated. Preferably a colonial pattern if you can find it. If you must use a quarter round, use as small as possible. I would probably just paint a small top beveled batten that is thick enough to fill the space between the floor and the cabinet side
One other thing I was thinking was to paint the existing wall trim (the one piece in the picture) the same color of the cabinet. Where it dead ends into the door casing would show the contrast.
I wouldn't, unless you change all the trim to match the kitchen cabinet color. The way the trim dead ends into the cabinet side creates the break and the trim style and cabinet style are different enough to signal to anyone who cares about this stuff that it was intentional. I know they are close in colour but once you start down that road, you wind up repainting all the trim.
Unless you have a lot of time and it really bothers you or your MIL I wouldn't go there.