Why doesn't it matter if a resistor is before or behind an LED wrt voltage drop?

One thing you need to remember is that voltage is relative. Voltage is a potential difference and it makes no sense to discuss voltage without a 'zero' reference.

In the case of your LED circuit there will be a voltage across the battery, a voltage across the LED, and a voltage across the resistor. If you add up all the voltages as you go around the loop, you get zero – up 9 at the battery, down 6 at the resistor, down 3 at the LED, the total is zero and you're back at the same point in the circuit. The LED only sees the difference in voltage between its two leads, as does the resistor. Since only the difference is important it makes no difference what order the parts are connected in.

As for current, it is only the same along a continuous path. Electrons are not created or destroyed (what goes in must come out). Since there is only one possible path for the electrons to take in your circuit, the current will be the same through all of the components. In a parallel circuit you see the opposite: all of the components have the same voltage, but the currents will be different.

If you want 3V across a LED it doesn't matter if you apply 13V to the anode and 10V to the cathode. There is still 3V across the LED.