its the drain line for water to drain
It should not have water in it, especially if freezing weather is expected. Drain the pool to below skimmer line and drain the filter and filtration equipment.
You should install a drain to prevent stagnant water from accumulating and hopefully prevent bacteria such as legionnaire or other water borne diseases and most importantly this drain should empty as an indirect waste as per modern codes
The drain from the washer to the main line is getting plugged and the floor drain is easier. Drain cleaner in the floor drain should take care of it.
Drain line is clogged.
Because both sides of your sink are connected to a common drain line, and the common drain line is slow.
I would say that if water had never backed up in your floor drain, then backed up even after you had the drain line snaked, Roto-Rooter didn't do their job. Once a drain line is cleaned, it should drain like it did before the stoppage. As for plugging the floor drain, it is there for a reason, I would not plug it. There are several items manufactured that will allow a floor drain to drain but if the line starts stopping up they will seal the drain off to prevent water from coming out of the drain.
Yes, be sure to pipe floor drain line properly into water closet piping.
The top hose.
Unless you know that there is one in the line before it gets to the main line, you should. Sewer gas can come up that drain just like any other.
u must have trash in your drain line,u will need a water ram to unclog