Alright, I installed the switching power supply and I get the startup sound every time now. The 4 led flash supposedly means it passed the rom/ram check. Right now I'm using the monitor hookup to power the switcher, so I have no idea if anything is showing on the screen or not. I check to see if it could play blind, but didn't have any luck. I also noticed that the second LED light on the PS was out; I guess that's why I was having problems.
I'm not sure how to properly feed power to the switcher and the monitor at the same time just yet. Once I figure out how to do that I can start troubleshooting what I believe to be a problem with the main board; the same problem that caused the never ending rug pattern. I read that it might be caused by a problem with the DRAM, but since I passed the rom/ram check I'm not so sure.
The monitor I have has apparently been capped, so I don't think I need to worry about it. Rebuilding the power supply sounds like a good idea; it isn't dangerous though is it?
One last question: The switching PS gives off a sort-of burning metal smell similar to a hot soldering iron; should I be concerned?