Quote "I don't remember a lot from DiffEq". Time to do a good review, isn't it? To keep it simple assume V = 0 and do the partial derivatives. What do you get?
The answer key is correct. Your answer makes no sense. ∂x/∂t = 0 is just an expression of the fact that x and t are independent variables. That's what's meant by a partial derivative. You keep x constant while deriving with respect to t. Well, if x is constant than its derivative with respect to...
No, it does not. The "right hand rule" is arbitrary you could just as well use the "left hand rule" and everything would work the same (with reversed direction for the magnetic field since its direction depends on the arbitrary choice of right hand rule. That's why the Magnetic field is a...