Should I install the bathroom cabinet before I tile the floor or after. The basement bathroom is about 5 - 10 and will only have 1 cabinet.

Tile first. That way the cabinet could be replaced with one of a different size without redoing the floor. It is also easier and saves cutting tiles and having to avoid tile slivers around the base.

You should install your tile first and then the cabinet. Someday down the road that cabinet may get replaced and who knows if the foot print will be identical.