Drywall – How to install a non-load bearing wall on a carpeted concrete floor and finished drywall ceiling

Looking at your wall photos, can I assume that seeing hinges on both left and right, the doors (2) meet in the middle? first we have to make some assumptions based on the yet unknown. if the walls are always dry, seasoned and pose no moisture or condensation issues, there is a couple of ways you can go ahead and finish he walls with drywall. i would consider putting a 1X6 up to a 1X10 finish pine board in the corners. This should clear the hinges. You can then install 1X3 strapping on the concrete wall with PL400 adhesive and short ramset nails or concrete screws. You can then decide if you want to install any blue moisture proof rigid insulation between the strappings. Then install a vapor barrier such as 4mil plastic over the strapping and then install drywall with 1 or 1 1/4 inch screws. You then can finish your drywall and build a neat trim at the ends onto the finish pine boards. The other alternative is to build a regular frame wall out of 2X3's or 2X4's and mount to floor and upper joists. This will allow more insulation and still allow doors to open almost fully, depending on the width of the pine trim at the corners. The wider the trim in the corners, the more fully the doors will open without being restricted by the new wall. If moisture on the walls is possible, you should really consider using Dry-loc or some suitable waterproofing before you even think about closing the concrete in behind a wall. Mold and mildew with ruin your good looking wall and potentially cause foul odors and a possible health hazard.

I would suggest that you mark where you bottom plate is, adjust circular saw to hardwood depth and cut it out (might need a jigsaw or dremel in corners).

Not exactly sure what type of wood flooring you have but I would do this in all cases. If your wood is floating you MUST do it.

If you have nail down I would still do it because you are introducing different amounts of pressure on the floor which could lead to buckling/damage over time. Keep the boards you cut out and they can be reinstalled. Also your drywall and baseboards will overlap the cutout points so you don't have to worry about how the hardwood will look after cut - you will have almost an inch to play with on each side.