Framing new wall/door
Framing like that looks fine and is standard.  Just put a single 2x flat at the top of the jamb instead of 2 and nail up into the cripples above it.

As for the screwing into the floor, the closer you can get a screw or nail into the floor the better.  We usually shoot some nails at an angle (toenail) at the bottom plate on each side of the opening.

Other than that get your measurements right and you should be good to go.