Why do we "need" resistors (I understand what they do, just not why...)?
Your understanding of how power flows through a circuit needs adjusting.

1. How much power flows through a circuit, and is taken from the battery or power source, depends on how much current flows through that circuit.

2. How much current flows though the circuit is dictated by how conductive the circuit is. If a circuit has a high resistance, it is less conductive, and less current/power flows. 

So, putting those two together and looking at your questions...


  1.What happens to the rest of the current not used (because of the resistor)? 


There is no "rest of the current" the current is defined by the resistance of the circuit.


  2.Does the LED then use ALL of the current available in the circuit? If not, where does the rest go? (Recycled back into power source?)


Again, the LED and its resistor define the current they will take. There is no "rest".


  3.Why does an LED "drop voltage" by a certain amount? And what happens to the rest of the components in series, does the voltage drop for every component, up until there is nothing left?


The LED has a more or less fixed forward voltage at a given current. The rest of the voltage is dropped across the resistor. That defines the current through the LED. 


  4.I recently watched a video, where the guy explaining resistors, drew a scetch showing 12v --> Resistor --> LED --- 0V (Do you choose your resistor to the extent of "using up all the current/voltage" before it gets to the end of the circuit? Youtube Video


In any series circuit, the applied voltage gets divided among the elements of that series circuit. The current is defined by what the circuit elements demand and is constant throughout the series circuit.

Keep in mind that voltage is simply a measurement of the potential for electrons to flow between two points.  It is always measured between two points, and a value of 0 volts tells us that there would be no current between those same two points.


  5.Why does a battery go into a dead short if you connect the terminals directly, but if you add a light bulb (resistor), it does not?


A dead short has virtually zero resistance and take a lot of current from the supply. A bulb has a resistance and takes much less current.


  6.I have done hours and hours of research, and I understand what a resistor does, but I do not understand why it is needed (to not dead short a battery? .. does this mean it "eats" all of the power before it returns to the Anode?)


Resistors are needed to set currents and adjust voltage levels through a series circuit. They are used for other functions too, like part of frequency filters, oscillators etc. etc.


  7.Why do different light bulbs work on the same battery (different resistance, but no dead short?)


Different light bulbs have different resistances.



In order to understand all this you need to familiarize yourself with Ohm's Law and Kirchoff's Voltage Law.



EDIT: Adding comment question since it's useful all on its own and may get migrated.


  Am I correct in stating the following: "If I put an LED directly on a 600maH power source, it will "use" everything that is available (600maH). Do I then calibrate for the resistor to resist enough current in order to feed the LED only what it needs?


A 600mAh power source is fairly meaning less here. mAh is a measure of how much charge and effectively total power a battery will supply in any given time. If your circuit takes 1mA the battery will last 600 hours. If your circuit takes 1A the battery will only last 36 minutes. Note the units... mA * Hours.

A bigger battery, of the same technology and voltage, has more mAh.

How much power it can deliver at any given time is dependent on the terminal resistance of the battery and how fast the chemistry inside the battery can react. A 3.7V 600mAh Li-Ion battery will deliver much more raw power than a 1.5V 600mAh Alkaline. Power and Energy is not the same thing. Ultimately, though, the load, the circuit, dictates how much it sucks out of the battery and how fast, assuming it's not drawing too fast, at which point the battery voltage will drop off. 

You have to think of a battery like the gas tank on your car. How quickly the gas goes down depends on how hard and fast you are driving. 600mAh only defines how big the "gas tank" is. The gas has to go from the tank to the engine through a pipe and the injectors. If you demand too much gas, it won't make it through those fast enough and the engine gets starved of gas.