Plumbing – install a sink that drains into this floor drain

Your vanity plan should have front side doors in line with where the sink is located. Then the drawers should be off to the left or right side where there is plenty of height and space toward the back to accommodate their placement.

There are multiple reasons to not put drawers in the vanity under the sink area:

1. People stand in front of the vanity at the sink. It is extremely inconvenient to have drawers opening out at knee height where the you want to stand.
2. You cannot place any sort of drawer directly in front of the sink and drain because there is just not enough depth from the front face of the vanity to accommodate the drawer. Any drawer placed below the bottom level of the sink would have cabinet face framing running across in front of the cabinet that would make it difficult to make access to the space under the sink.
3. Having drawers in the space under the sink will restrict access to the water shutoff valves keep it from being possible to quickly inspect the drain area in the case of suspected leaks.

So I really recommend that you not put drawers in this part of the vanity cabinet. You will find plenty of things to store in the area under the sink behind the doors.

There is probably soap and hair plugging the drain lower down. a small hand auger might be the best bet to clean it out. Chemical drain cleaners partially open the drain then they plug up again much quicker and you end up spending more on chemicals than the cost of the auger.