If i were to ask my mom where my towel is I would say: " Vart är min handduk?". A "duk" is a thing that you put on a table, I would never say "Vart är min duk?" wanting a towel "Duk" is not the same fabric as a "handuk" so to answer the question no "duk" does not mean towel. Det var så lite så! ( No problem)
I think "duk" is cognate with English "duck", in the sense of "duck cloth", not the waterfowl or the verb. It goes back to an old Germanic root meaning cloth, so woven metal is not surprising.
It's a bit of a grey area :) but generally speaking, duken blir smutsig can be used in much the same way a continuous sense is used. So it's more like "the tablecloth is getting dirty" in that case, which probably corresponds to what your friends mean. Then again, it's hard to tell for sure without having been there. :)