In the first scene that we see House with his new leather jacket, the lettering on the right side of the jacket spells "KTAI Sports". In a scene a few minutes later, with House and Wilson standing by House's new motorcycle, the lettering on the jacket reads "RTAI Sports".
When House is talking to his parents about his motorcycle and his dad says "You know what your problem is?" House replies "Shifting gears?". Motorcycles have the brakes on the right (handlebar is front brake, foot is rear brake). Shifting gears is done with the left foot. Since he has a problem with his right leg, it's braking that's difficult, not shifting gears.
The improvised, single point cane holder on House's motorbike is far too flimsy. You can see the grip end of the cane wobbling even at a standstill. He'd lose the cane riding across the first couple of fast bumps.
In the diagnosing scene, with the phone call from House's mom, he says: "that begs the question" when his meaning is "that raises the question". House would not mix up these very different sentences.