REX647:
Since I intend to operate the MOSFETs via the Arduino, I should step the 12V down to the 5V operating voltage of the Arduino Pro Mini. I probably won't need to power the Arduino through the RAW pin then, correct?
If you provide a step-down regulator for 5 V, then you supply the Pro Mini (I like Pro Minis!
) at its actual operating voltage via the 5 V pin and the "Raw" pin is no longer relevant.
On the other hand, given that you are not attempting to power any other components or modules (apart from a few 470 Ohm resistors), the on-board regulator should be able to handle a 12 V input to "Raw". Just feel the regulator - if it does not burn your finger, it is OK.
REX647:
You say "and the other components" after you say stepping the voltage down to 5V on the Arduino. Do you mean the LED strip as well? Doesn't this need 12V to operate though?
Given that the LEDs are operating from the 12 V, and unless you add other things which do require 5 V, I see no need at this point to provide another 5 V supply.
REX647:
You say that the 12V/2A power supply is the best choice. I will get this along with the other components I need to get (when I get the MOSFETs for example).
OK.
REX647:
With respect to the MOSFETs I listed, are you saying they're a bit overkill for this sort of application? If yes, what might be a more appropriate component?
Overkill is in this case, very good. They will not heat up appreciably and thus will not require a heatsink. The only reason to use a different component would be if it was important that it be substantially cheaper. Most of us buy components from eBay unless we are in a hurry. 
REX647:
So if I were to think about the MOSFET pinout with respect to the transistors used in my original plan, I should attach the Arduino control to the leftmost pin of the MOSFET (gate), the ground to the rightmost pin of the MOSFET (source), and the line to the LED strip attaches to the centre pin of the MOSFET (drain).
If you have identified the pinout of the FET correctly, that should be it.
REX647:
This happens three different times for each of the three colour channels, of course. I will leave the 470 ohm resistor inline with the gate of each line. When you say "10K resistor directly from that Arduino pin to ground," which place are you referring to? From the GND next to RAW to the power supply, or do you mean from the MOSFETs to the ground of the power supply?
You will want a 10k resistor to ground from each Pro Mini pin which also has a 470 Ohm resistor to a FET gate. This ensures that whenever the Pro Mini is not controlling it as an OUTPUT, it will switch the FET off. A common mistake is to connect the resistor between the gate of the FET itself and source, but this would then form a minor voltage divider with the 470 Ohm resistor and albeit only slightly, reduce the voltage switching the FET on.
REX647:
(Perhaps the 10K ohm resistor doesn't even matter, as I plan to plug the whole system into the wall outlet and leave it on indefinitely. A little bit of initial flickering or flashing is probably okay if it stabilises itself after a brief period of time.)
Maybe, maybe not. The flickering or flashing may not annoy you, but if the voltage on the gate spends any time in a mid-way state, neither switching it fully on or fully off, the FET will be both passing current and have a non-trivial voltage across it and will then heat up.