Could my memory problems and rls 5 out of 7 days not just nightime, also hands feet and lower leg paid, have a common cause?

Yes: Poor sleep related to restless leg syndrome can result in poor memory.

Yes: Many possibilities could be considered, and incidentally, even ms could be considered. However, might consider CNS sarcoidosis, a vasculitis, an infectious process (hiv). Maybe with the limb problems, part of this may be a small fibre neuropathy, and if biopsy proven, could narrow the diagnostic choices to about 9 entities.