What kind of filaments can I use? This is probably the most basic of questions. When I was shopping for a printer I saw models that would list a slew of certain plastics that the printer can print while other printers (different suppliers) would only list PLA/ABS. At the same time, the instruction manual that came with my printer only listed PLA and ABS.

What is actually used to determine the possible material (other than diameter)? I assume that if the nozzle can reach the required temperature to melt the plastic and the bed can reach the necessary temperature to maintain bed adhesion, then my printer can print that plastic. Am I correct? 

For example, my nozzle can reach 250 °C, but the bed can barely reach 80 °C so I can print with ABS but not with ASA (since it needs at least 90 °C on the bed. However, if I switch out the heating element for something beefier then that opens up anything that needs a hotter bed. Right?

Let's assume that I have the means to change nozzles and extruder type to fit the need of the filament. The goal of this question is to limit to one or two limiting factors of the printer without major modification.
There are other factors besides temperature. Certain "soft" filaments won't go well through all extruders, and some other filaments with wood/metal particle additives don't go well through every extruder type and can cause clogging. Even people who do tend to print a lot of these materials will often use a specific nozzle just for those filaments.

That said, you still might be able to print even with something like ASA that seems unsupported. You can do this by taking additional measures for bed adhesion, or upgrading or swapping your nozzle or hot end.

If you can print PLA and ABS, you can probably also print so-called PLA+ and PETG, at a minimum.