1- Yes, but the more it unfolds, the thinner and weaker the part of it that reaches the object will be. At one point it may be thinner than an atom, at which points further questions become too complicated for me to bother trying to answer. If Plank’s distance is mentioned I will run away.
2- If it goes into the bath water and you consider the water to be a continuous medium, then the surface of water touching it will also be infinite. If you consider a scale too small for the water to be considered a continuous medium, however, I will leap out the window.
The problem with washing it is more with trying to scrub it then just submerging it in water. But as you pointed out it probably gets very brittle further out so you might hurt it if you try to scrub it
