For the problem of the 1 pt relay (N) being locked on, one of these highlighted switches is probably stuck closed, or the tabs on the back of the switches are shorted, or the N/C switch on the ones score reel that should open when the score reel plunger is pulled in, isn't opening (that's the "On Add Units Unit" switch).
For the starting problem, as Howard mentioned, make sure there are credits on the machine, and if there are, make sure the "Zero Position Replay Unit" switch (in red below) is closed - it should only be open when there are zero credits. The red part of the circuit is the only part that's different between starting the machine with the coin chute and starting by pressing the Replay button, so the problem must be one of those 3 switches.
Egghead start circuit (resized).JPG
Is your score motor turning continuously at startup? Is that why your bank reset coils keep firing?