The crackling you heard (like tiny bacon frying in a tiny pan) is the HV generator firing up. The MOSFET is getting hot because the circuit stopped again with the gate held high. It's definitely around the crystal circuit.
Please mount the crystal and the load caps fairly tight to the board (but not so tight that they are under physical stress) , pull out the controller and re-seat, after checking that the contacts on pin 9 and 10 are clean. It's a kit that has been hanging around for a while and there's a chance that the contacts are tarnished.
Well, after removing Q2 it looked like the pads on the pcb came out too. I was able to re-solder one leg of Q2 and have continuity to C5 and pin 9 of the ATMEGA. I can't get the other leg of Q2 to solder to the board. Can I just run an external wire from that leg of Q2 to C6 and to pin 10 of the ATMEGA?
It's a bit fiddly, but in the interests of getting on, you can try.
I better solution will be that I send you a small kit of parts so you can have another run at it. It shouldn't have been been such a painful experience, and to be honest, I'm not sure we can totally rule out faulty components.
Well, I patched the crystal to determine whether I have one or more bad components or just a bad soldering job, and I get the same results as before - all white LEDs, but touching the crystal causes them to change colors. So at this point I need to replace at least some of the components - should I replace the crystal and the C5-C8 capacitors? Anything else I should check? Or perhaps better to take Ian up on his offer?
Truthfully I've learned a lot more having to troubleshoot than I would have just following the instructions, so I wouldn't consider it too painful.