The diagram shows first four resistors in series then four resistors in parallel.
For the resistors in series the current flowing into the wire, $I_{in}$ must be the same as the current flowing out, $I_{out}$ because the current can't escape from the wire. There is only one route for the current to flow through the wire so the current has to pass through all the resistors in turn. That's why the current passing through every resistor must be the same.
Now look at the resistors in parallel. The point here is that the top ends of the resistors are all connected together so they must all be at the same voltage $V_{in}$. Likewise the bottom ends are all connected together so they must be at the same voltage $V_{out}$. That means all the resistors have the same voltage drop across them of $V_{in} - V_{out}$.
In regard to the first, we have a 5v supply, which from my
understanding means that if you were able to enclose a coulomb of
charge eminating from the negative terminal, you'd find that it has 5
joules of energy.
This isn't the typical understanding. A 5V (ideal) supply (source) maintains a 5V potential difference across the terminals independent of the current through.
It follows that 5 Joules of work is done by the source on 1 coulomb of (positive) charge in moving through the source from the more negative terminal to the more positive terminal.
Conversely, 5 Joules of work is done on the source by 1 coulomb of charge moving through the source from the more positive terminal to the more negative terminal.
1) Why is less energy lost going through R3 than R1? or rather, why
does an electron have more energy after passing through R3 than an
electron passing through R1
There is 5V across R1 but only 2.5V across R3. Why? There twice the current through R1 than through R3.
If the electron loses all its energy on exiting R1
It doesn't; the (average) kinetic energy of the electrons entering is the same as the electrons exiting (the current entering is the same as the current exiting).
However, the exiting electrons have less potential energy.
Why doesn't resistance value have an effect on the total energy loss
of an electron passing through the circuit?
The resistance has an effect on the power (rate at which work is done); the electrons must lose the same amount of potential energy in travelling through the resistor(s) (all else being equal) but the number of electrons per second is reduced when the resistance is increased.
Best Answer
The energy loss of a specific charge (like an electron) on passing through a resistor depends not on the resistance, but on the voltage difference. That voltage difference itself depends on the current.
If the path through one of the resistors wasn't removing all the available energy from the charges, they would continue to be accelerated, increasing the current. That increasing current would raise the voltage difference in the resistor, draining more energy.
The process is a feedback loop that converges to a steady-state where the voltage drop across both resistors is equal to the voltage gain from the battery. Because of the differing resistances, the currents through $R_1$ and $R_2$ will be different, but the voltage drop will be the same.