Does anyone of you had access to solar panels? I bet not.

How do I connect a 6V source to a 12V fan? Do I need to add a resistor?
What's going to happen to the fan if I supply only 6V? Does it spin at half the speed?
What if I connect two fans in parallel, giving each fan half of the current? Do they spin at half the speed or simply not work at all?

You must look at the datasheet from the fan manufacturer. They often specify starting voltage. Most of the time it's about 1/3, maglev fans go to 1/4. Some datasheets have tables to give you an idea of how current/voltage relates to rotation speed.
A 12V fan does not require a 12V input to function. It requires 12V to function at its nominal conditions. A simple way to make your 12 cm case fan more silent is to just switch its connectors from the 12V rail to the 5V rail. It will still work.

Since V=I * R, just adjust with the appropriate in-line resistance to get 12V across the fan

This is not required as silicon will generate 12V by using two panels 6+6V. Anyway, being both I and V variable, we're going to have some fun calibrating them: just consider nominal panel voltage.

I've pumped well in excess of 50W in a 8W fan (produced by three 22 Wp panels). I have experienced some overheating and shortened lifespan but I hardly believe there's any chance of fire.
There's no way this amount of continuous current could give anyone "a nasty shock". Maybe only if you put the terminals in your mouth.

As a side note, due to a cabling failure in my solar array, I discharged over 4000W of power (at ~200V if memory serves). It hurt, yes, but nothing really nasty, it didn't even burnt my skin.

So when looking at an electronic device with the V A ratings (e.g. 12V 160mA), if the V can be variable, is the amperage the one to look at when it comes to measuring how much power you need?

For a generic electronic device I'm afraid you can only guess how it is going to work. They are incredibly complex, sometimes using multiple voltage rails internally, with the ability of switching off whole logic blocks. The wattage is only nominal, it will often be lower.
DC fans are not "generic electronics". In my experience they work more or less as the datasheet suggests. That is, given nominal voltage and a current they will rotate at the specified speed. You might try do that in reverse.

I need to make sure that my power supply is able to supply all four fans. All four fans combined, if connected in parallel, require 640mA. If I only use two solar panels to power four parallel fans, it will cause voltage drop, and may then damage the solar panels, overheat, or catch fire.

Solar panels have an internal maximum current. It's fairly high. The idea is that they should be able to tolerate their own current when not discharged (that is, you don't need an absorber to make sure their juice is used). Cell quality is key factor in understanding if the panel can be damaged by current not being absorbed. FYI:

• Motech cells: I discharged the production of 6x2 cells on two serially connected panels in a reversed polarity 6x2 panel. One cell got an hotspot. Since then the damaged panel had reduced performance (about 70%).
  
• Bosch cells: tried to cause a extreme partial shadowing effect, four corner cells in a 240Wp panel were screened. No problem observed nor measured.
  

Keep in mind that in my experience solar panels works most of the time at about 60% nominal power. You probably need more than 4 if current is your goal. Yes, you can trust voltage. The cells will pull out usable juice only after a certain voltage is reached. After that value is reached it will stay more or less there and extra irradiation becomes current.

I have no idea how the panels you have selected are going to react. They don't look very high quality. If you need 8 of those, you might want to check out "commercial grade" panels instead.