Mine also does the explosion when I turn it on, so I dunno that I'd worry about that. If you don't have the tools to troubleshoot though, it's gonna be pretty tough to figure this one out. The schematics are pretty good, and if you get the explosion sound, you know the output stages and speaker are good (although the static could be coming from there).
If you cut W11, that will isolate the output stage and I believe background, laser, and star... if there's no static, then everything on the output is likely good (you can touch your finger to it and see if you hear some humming), although the background, laser and star sounds come in at pin 2 on IC27, which may or may not work. If you still have static, you could remove the input at pin 2 of IC27 and see if it goes away.
Then, W6 through W10 connect each individual sound output to the output stage, so you could cut each of those individually to see which are causing the static, and which work (of course reconnect W11).
From a quick look, there's really nothing common on the analog side between all except explosion... the only thing I see is IC3 (74LS377) controls the enables for all except Laser, Soft Explosion, and Loud Explosion. Other than that, IC20 - IC22 are the opamps for all those sounds (IC18 and IC19 are for soft and loud explosion), so you could check them (but it seems unlikely that they're all bad).
Anyway, I hope this helps you get started, although troubleshooting would be MUCH easier if you had a scope, and although I don't know how much a working board costs, it may be worthwhile to just buy a known working one, or send that out for repair.
DogP