Question

Why would electrons leaving a resistor have more potential energy than when entering?

Answer 1

Usually they would not. If the resistor were a coil in a transformer, however, they would leave at a higher voltage (potential) than they entered.

Answer 2

Yes. In fact remember that electric potential is given by:\[U=\frac{ 1 }{ 4\pi \epsilon 0}\frac{ q _{1}q _{2} }{ r }\]Now remember that electrons flow toward the positive terminal in a circuit, this means that q1*q2 is going to be negative so as r gets smaller, the potential gets even more negative, meaning you'd have to put more energy into the system to separate the electron from the positive terminal. The closer the electron gets to the positive terminal, the more energy you need to separate the two.