No pitcher is not able to fake to first unless he steps off the pitching rubber first. You probably won't see that play because if a lefty steps off, the runners get back. Any attempt at a pick off to first by a pitcher righty or lefty, the pitcher has to throw the ball to first so that stops any fake plays.
Most of the time, pick offs to third aren't used because it is too risky. If you overthrow, that's a run. Plus, very few runners are trying to steal home so there is no need to hold the runner too close. But a lot more are trying to steal second so that's why you see so many pick offs there. Plus, any errors and the runner usually only gets to second.