Confusion about relationship between energy per coulomb, and the amount of coulombs in current
electricity in a current is the movement of free electrons through a conductive material.

This is one kind of current. But it's also possible to have positive current carriers. Batteries and many kinds of transistors would not work if electrons were the only kind of current carrier.

Voltage and Amps are directly proportional.

This is true in a linearly resistive material. For example in a copper wire or a resistor.
In other components, there will be a different relationship. For example, in a (junction) diode the current goes up exponentially with increased voltage across the device. In a transistor the current through one branch of the device may depend on the voltage across a different branch of the device.

Which of the following is correct?

Again, it depends what kind of device you are investigating.

More couloumbs AND more energy per coulomb:

This is the behavior of a resistor. If you increase the voltage across the resistor, by definition, you've increased the energy per coulomb lost as charge flows through the device. And, because of the nature of the resistor, you will also increase the current (rate of charge flowing through).

More coulombs BUT same energy per coulomb:

This would be the behavior of a voltage source when you change its load. You can increase or decrease the current you draw from it, but it will always (to the extent it's an ideal voltage source) produce the same potential across its terminals.

Same number of coulombs BUT more energy per coulomb

This is the behavior of a current source. You can apply whatever potential you want across it, and it will always (again, in the ideal case) produce the same current.