Low outlet placement and baseboard issue
It's possible to do, but not easy. There isn't much surface area on quarter rounds. Usually there is a significant gap between the floor and the wall to allow for expansion. Usually, the quarter round is nailed to the baseboard, which is nailed to the studs. Without baseboards, the gap between the flooring and the wallboard doesn't leave much material to attach the quarter round to.

Just because the previous baseboards looked bad doesn't mean your new ones have to. Since it sounds like you're pulling the old ones out, you have the opportunity to select ones that will make installation around the outlets easier or prettier.

Alternatively, if all of your electrical wiring in the area comes down from the ceiling, it might be a simple matter to install new electrical boxes at an appropriate height and move the outlets up.