Why does a resistor need to be on the anode of an LED

Electrons have a negative charge. Current is Coulombs per second. Coulombs are positive, so a coulomb moving one direction is actually caused by electrons moving the other direction in meta.

When we discuss current we are discussing the flow of positive charge particles. If the flow of current is actually made up of negative particles flowing in the opposite direction it makes no difference, that is two negatives which cancel. It is just a case of math and sign convention.

The only time you pay attention to the actual carriers is in something like a semiconductor where you need to know what is happening as you travel from electron carriers in the conduction band "holes" in the carrier band. The holes are positive charge carriers, but that is because we are counting the absence of an electron, the actual current is still made up of many many electrons slowly drifting.

Is current always electrons?

Actually if you ever model electrical systems in the body you will find that you can accurately model a neuron using a transistor network and such. Much of the current relates to ions like potassium. This means that you really do have motion of positive charge articles. It still is just drawn as a schematic because it does not matter what the charge carrier is as long as your schematic models the electrical properties well.

Is the electron moving the power?

Often people think that your power you are sending is the electron. In reality you are sending electromagnetic signals. You can slow the speed of your signal(ie. the power) propagating down a long pair of wires(one the signal and one the signal return) by changing the dielectric between them. This means two unshielded copper wires just sitting in space will actually have their signal travel near the speed of light. Your coax cable will probably travel very close to two thirds the speed of light. The electrons drifting are a function of the electric field that is present. If you were to measure how fast the electrons are drifing you would find it on the order of a few meters per second.

Electrons are flowing from negative to positive when there is a connection for them to flow. You can also think of it as "holes" (or the absence of electrons in an atom) flowing from positive to negative.

Because of historic reasons, we generally think of conventional current, or the "holes", rather than the actual electron current. You are correct that the positive terminal just "pulls negatively charged electrons."

There is also no magical node called "ground." We simply assign a point that is easy to keep track of to refer to as ground. In a circuit, you could refer to the positive terminal of the battery as ground, and simply be working with negative voltages (though it would likely cause some confusion for others interpreting your work).

Likely in the case of shorting the battery to the chassis of the car, the negative terminal of the battery is already connected to the chassis. Possibly this is done for some sort of EM shielding (vehicles are inherently very noisy environments). When you make another connection from the negative terminal of the battery to the chassis, no current flows. When you connect the positive terminal, current flows through the chassis of the vehicle to the negative terminal.