Character mistake: Sheldon is wrong about the physics of pushing the furniture up the stairs. This is not just a question of work (energy) but also of power (energy/time). Since he and Leonard cannot produce an unlimited amount of energy per second, they may need to push it slowly, contrary to what he asserts. This is a very serious and very basic error for Sheldon considering he has a PhD in Physics.
Character mistake: Sheldon says (during Fun With Flags) "But like the flag over Fort Sumter, I'm still here." The Sumter flag was lowered when the fort surrendered to the Confederates. The flag over Fort McHenry was never taken down despite the British bombardment and inspired Francis Scott Key to write the National Anthem about it. Whouldn't a vexologist know that?