Changing the caps often helps in reset problems.

In a battery-backed up CMOS RAM, the current is so low that the usually stated 0.7V drop across a diode is actually much less, about 0.2V even in silicon diodes. Anyway you should be able to use three batteries.

There was a modification for the reset circuit in an old Gottlieb service bulletin. You can find the description in
http://tuukan.fliput.net/sys1_en.html#viat

Looks like you have a later revision board that already has the modified reset circuitry. Next step would be to replace Z2, and also check transistors Q5 and Q6.

One common cause for all solenoids on is bad or missing -12V from power supply. So if your power still has the original electrolytic capacitors, they should be replaced, at least C1 and C4.

Game starting correctly only after turning power off and back on is a classic symptom of bad electrolytic capacitor. Replace C1 on power supply. You can also try warming the cap with a hairdryer before turning on, and see if it then starts correctly. Failing caps work a little better when they are warm.