I'd add a back band around the outside edge of the door trim. It can be fancy profile or a simple piece of 3/4" by 1" (or as needed) rectangular trim. Here are a couple of visuals:

If the setback for the locks is normal (2-3/8" to 2-3/4" from the center of the knob to the door edge), and you're pinching your fingers, it probably means that the trim around the door is fatter than normal, or the door is weirdly trimmed out. Hard to say without a picture.

I think a possible solution for you would be to lay the new flooring as close as possible to the molding, then scribe the outline of the molding onto each piece. Cut along this line at an angle so that the top of the flooring is a bit longer than the underside, then lay the floor as close as you can to the molding, say 1/16 inch. Fill the small gap with colored wood filler which matches the floor. They also make wax sticks which serve the same purpose. Your gaps should hardly be noticeable that way--who goes around on hands and knees inspecting your wall/floor transitions!? Yes, flooring manufacturers suggest a wider gap, but interior conditions are fairly stable, and as long as the two surfaces aren't actually butted tightly, I don't think you will have a problem, especially with the angled cut. Have you decided yet on solid wood flooring or on laminated pre-finished?

It is too small for either of those walls.Could you post a pic of the whole space those sofas seem too large and crammed into a tiny area.