Heat and small appliances
1.When you plug an appliance into a 120 V wall outlet, some amount of amperage will run through the wire depending on the amount of resistance in the circuit (I=V/R, where the voltage is a constant 120 V). Resistance in the circuit depends on the material resistivity and the length/shape of both the wire and appliance.


Correct.


  2.The difference between a high amperage appliance (like a refrigerator or space heater) and a low amperage appliance (like a light bulb) is the resistance in the circuit. A light bulb draws less amperage because there is more resistance. In other words, resistance is used to control the amperage drawn by an appliance. The appliance is intentionally made to have the right amount of resistance so as to draw the right amperage. 
  •Is this assumption correct?


Yes. A little more complex than that, but essentially correct.


  3.More resistance creates more heat. 


Incorrect. For a fixed input voltage, LESS resistance creates more heat. 

\$Power = Volts^2 / Resistance\$

As the resistance falls you consume more current and hence more power.


  This is due to the electrons bumping into the atoms in the material they are moving through. I kind of envision this as being like having more friction, so therefore more heat. This is the reason a frayed wire can heat up and cause an electrical fire.


True, but again, it is dependent on how fast and how many electrons are moving. Higher current = more collisions = more heat.


  •Does this mean that since a light bulb has more resistance than a space heater, it is more likely that it can heat up and cause an electrical fire? Are small appliances therefore more dangerous than large appliances due to their higher resistance?


Your invalid assumptions make this a little invalid.

Further, temperature change is also dependent on geometry. The filament in a 100W light bulb gets orders of magnitudes hotter than your 750W refrigerator because the heat is concentrated in a small area. It is important to separate hot from heat here. Your fridge puts out more heat than the bulb, but does not get so hot.


  •Does current in itself create heat? 


We already covered that.


  So when you reduce resistance and therefore increase current, does more heat get produced (although heat due to resistance decreases)? Conversely, does increasing resistance (e.g. fraying a wire) also help it cool down since current is reduced?


Again you got the resistance part backwards. 

Fraying a wire is a little more complex. What you end up doing here is increasing the voltage drop across that frayed part in the wire, which will have a higher resistance than the rest of the wire, adding another load1 in series with the appliance. This new load steals some voltage from the appliance. The total current is reduced a little. That voltage drop times whatever current the combination load continues to take generates heat in the frayed part. Since the frayed part is small, that heat turns into HOT. If it is frayed enough it can actually start a fire.







simulate this circuit &ndash; Schematic created using CircuitLab



ADDENDUM

1 The term "LOAD" can be confusing in EE. A "LOAD" is generally defined as something that consumes power. However, when you add resistive loads in series to a fixed voltage supply, the "load" on the supply goes down not up. Only when you add loads in parallel does the load on the supply go up.