Your monitor is missing one axis of deflection until it warms up? I assume it's a G05, haven't looked at your pics, but I'd put money on a bad solder joint somewhere on the deflection board,
Bad solder joint would make sense. Could it be some capacitor that is dead and does not allow enough current to supply the part that is responsible for the horizontal deflection? Will take apart the boards as much as possible and will check the transistors and diodes as well.
Btw, the monitor looks to be Sylvania type 15st4730r, part number 17-6152-05, some EIA 312.
Another question, if I work by the tube for a while (exposed neck, etc.), how much electro-magnetic radiation does this produce? Even if it's turned off, when I get really close to the tube my eyes start to hurt a bit and I'm getting kinda of a headache. I didn't feel this for at least two years since last time I spent many long hours sitting at a 19" crt computer monitor (not to mention "healthy" monitor glow in my skin

). With lcd I don't feel this.
For the sound issue, look carefully around the battery for any signs of leakage or previous leakage and corrosion. These boards are known for acid damage and it can cause all kinds of odd problems.
The battery leaked and was removed about 2-3 years ago. There is quite a bit of corrosion. What bothers me a bit is that the aluminum standoff piece is touching part of the exposed metal circuit where acid was. The damage is mainly just at the edge. Not sure, if it causes short somewhere. That part seems to be "-" for the battery. Right now the game does not save high scores. Many of the chips were replaced by ArcadeWorks. There is even a tag on the daughterboard.
Congrats on owning an Omega Race! Wasn't a very popular game when it was new, but I always enjoyed it. Have an upright myself. But then I've got a soft spot for any vector.. 
D
Thanks! Actually, Omega Race was top 10 money maker during 1982-1983 time period.

Not too shabby. With a run of around 35,000, it had to be a pretty popular game. I believe Asteroids was 50,000? With many games, cocktail version was coming out later on. I believe Omega Race cocktail came out in 1982. Looking at the manual some of the drawings are from second half of 1981. Pictures show the death ship as a star (or a pentagram

).
http://www.sendspace.com/file/duiksd - acid burn
http://www.sendspace.com/file/ldacyu - pretty new looking mobo
http://www.sendspace.com/file/0nxvdt - table # 1436 (why does it say "regular"?)
http://www.sendspace.com/file/htwh59 - tube
http://www.sendspace.com/file/z8ktdj - closeup 1
http://www.sendspace.com/file/oq3hgr - closeup 2