Ok, in response to Ian, I've tested continuity on the ATMEGA from pins 8-9, 9-10, and 10-11. No shorts there.
I also checked for continuity between the legs of the capacitors at C5, C6, C7, and C8. All showed open except C7, which starts at 0 and runs up to about 13k ohms. That seems suspicious, because the other 100nF capacitor at C8 shows open. Could that just be because I'm measuring it on the board, or could it indicate an issue with that cap?
I'm also uploading some photos of the crystal area. I will go in this afternoon and clean up the board per Ty's comments. Upon closer inspection with a flashlight it looks like there is flux residue remaining on the back of the board that I'll clean up as well.
I cut down all of the tails on the back of the pcb and thoroughly cleaned it removing all flux residue. I then applied power again.
-Initially, all white leds as before.
-When I put my finger on the crystal, leds dimmed to red (with some flashes of blue) as before.
-Upon running my finger along the crystal, all the leds turned bright red. There was some faint crackling/buzzing (not sure from which component), and the neon colon tube flashed partially on/off/on. I then checked the MOSFET and it was VERY HOT, so I disconnected the power.
Based on the hot MOSFET, I went back over every component in the HV circuit. All are in the correct locations, and no legs are touching/shorting. C1, C3, and C4 are all correct polarity. D1 and D2 are in place the right way around, but it looks like both are UF4007 (I can tell for sure that D2 is, the markings are on the underside of D1 but I know it starts with 'U' and ends with '7').
I also checked over the back of the board to make sure I didn't have any solder joints touching/shorting, and I don't.
The only thing I can think that may be wrong is a bad solder joint on the crystal. I may pull it off the board and then resolder, but wanted to post all of this here first in case anyone has any better ideas.
The crystal connections are a possibility. It's also possible that the crystal itself is marginal. Also it's possible C5 and/or C6 are bad or have bad connections. They look OK from your pictures, but sometimes things are not as they appear. If you have a solder sucker, try removing and re-soldering those components.
I'm with Jim on this: it is almost definitely something to do with the crystal and its loading capacitors. If the crystal circuit is not oscillating the controller will not run. If the crystal circuit is oscillating unstably the controller may run a bit but it will be unstable and do weird things which could certainly include firing the MOSFET erratically leading to it overheating. Crackling sounds would result from erratic firing too.
Look into it later when the dust is clearing off the crater.