The tap is probably next to the bath (in such a way that it's outside the frame) and can be swiveled over the bath. Allows the user to fill a bucket or the bath easily.
The strange lid type thing is in fact a lid for the tub. I assume they're expected to keep dust and debris out--especially if the family is planning on reusing the water rather than discarding it after each use. Japanese use the tub for soaking (like a hot tub) rather than for washing, so it's expected to be reasonably clean for the next family member.