you don't actually have to follow the inside to outside ram pattern, for instance if slots 1 and 2 are burnt, you can not use those slots and instead put them in 3 and 4 or in slots 5 and 6. Like for instance I had ram slots 5 and 6 burn out and can put them in 7 and 8 if you didn't already max out the ram. It should also boot if you just put ram in only slots 7 and 8 or even 3 and 4
You can desolder and resolder the slots, that would be the logical way to fix the problem unless you have zero soldering experience in which you'll probably do more damage than good there are tons of little transistors all around the ram slot pins.
eBay is going to be your cheapest route in finding a new logicboard, should run like $40-$70, I saw a dual 2.0 (june 2004) logicboard for like $60 or $70 the other day, you just have to make sure you have the right year and model, the only boards that are universal are the ones that are over the 1ghz frontside bus, so (dual 2.5 (1.25ghz), dual 2.3 (1.15ghz) and dual 2.7 (1.35ghz) can all use the same logicboard and the cpu plugged in determines what the bus runs at, where as the dual 2.0 has the 1ghz fsb and the dual 1.8 has the 900mhz fsb and only run at that. You can tell the difference between the 2003 and 2004 boards by the serial sticker on it to verify what you need 2003 had the old sticker 2004 had the newer sticker, I'll post a picture in a minute showing you the difference.
All of them will have the same solder issue because they used the same solder throughout the entire process silver bearing solder with no lead, and there's your reliablity problem,that why the nintendo from 1981 still works like the day it did, because it had lead in the solder, though it appears they used some sort of glue in it because of the yellow rust like color on it and around it.