Drive the flat edge of a nail puller behind one end of each existing baseboard with your hammer. Place a piece of scrap lumber between the nail puller and the wall to protect the drywall surface. Pry each baseboard away from the wall and remove it. Take your time to avoid damaging the wall or the baseboard. Remove the nails from each piece of baseboard and save for a future project.
Paint the ceiling and walls with oil-based enamel. Apply the paint around the edges with a paintbrush and fill in the remaining areas with a paint roller. Paint each of the window and door frames with the same paint using a paintbrush. Let the first coat of enamel paint dry overnight and apply a second coat to protect the entire room from the dirt and moisture that accumulates in a mudroom.
Remove any carpet from the work area. Leave any other flooring in place for extra protection. Unroll a sheet of vinyl flooring. Lay lumber flat on the flooring and leave it overnight to flatten the sheet. Remove the lumber and lay the vinyl on top of the existing flooring. Cut the vinyl with a razor knife to fit the mudroom. Remove the vinyl and set it aside.
Cover the floor with vinyl adhesive using a toothed tile trowel. Lay the vinyl flooring over the adhesive. Flatten the flooring with a wallpaper roller to remove lumps and air pockets. Let the adhesive dry for 24 hours before continuing.
Unroll the rubber baseboard. Cover it with lumber and leave it overnight to remove any curls. Remove the lumber. Cut the rubber baseboard to fit the perimeter of the walls with tin snips. Coat the back of the rubber baseboard with construction adhesive. Press it against the wall. Stack lumber against the rubber baseboard and leave it overnight while the adhesive cures. Remove the lumber.
Cover the seams at the top and bottom of the baseboard with a bead of caulk. Place heavy-duty welcome mats beside and in front of each entrance to have a place to clean and leave your shoes.