I believe that Sheldon is pragmatic like an 80 year old confused by inflation, insisting on paying what he thinks is the correct amount of money to live, and not real market value. Of course, Sheldon is not confused by inflation, but his Aspergers (which didn't exist when his mother had him tested.) sees through the changing human condition.
I'm actually amazed that Sheldon doesn't own the building, and hasn't mentioned it because the topic hasn't come up... And yes even if he owned the building for tax reasons he would still pay rent to himself.
For a time his flat's wyfy pass word was "Pennyisafreeloader."