If I place a compass over a wire(such that the wire is positioned north-south) with charge flowing through it, and it points northeast, how can I determine the direction of the electron current flowing through the wire?
A mnemonic memory aid is the right hand rule. Don't grab the wire for real because if you accidentally ground yourself, you will likely get a shock.
Stick your thumb out on your right hand, and imagine holding the wire with the wire running north and south. This can with either your thumb pointing north or you can do this with your thumb pointing south. As your fingers curl around the wire so are the magnetic field lines circling the conductor. with the current flowing in the direction your thumb is pointing. Think of your fingers as start at north and wrap around to south
The compass will give different results when placed above and below the wire. That is because above the wire the magnetic field points in one direction, and below the wire the magnetic field points in the other direction. Just as below the wire your fingers point one direction, but as they curve around to the top they are pointing in a different direction.
If your thumb is pointing south as your fingers curl around the top of the wire, pointing to the wires magnetic south, you will find that the wire's magnetic north is in your east. That is, when current is flowing south in your wire, and the compass placed on top, it will be deflect to the east some. The amount of deflection will be related to the induced magnetic field strength.