the red button relay powers when the game is turned on because the anti-cheat relay switch is closed at that time (anti-cheat switches are drawn in the just-turned-game-on state - i.e. before the relay powers forever on the first cycle).

the logic behind it is that when you turn the game on, the game should reset when cycled ... even if you hold in the yellow button and drop in a coin. Ditto if the tilt trip relay tripped. With the RBR powered, the shutter will open if it's closed.

if you have strapped down the anti-cheat relay armature so the credits don't get removed between power cycles, then the red button relay won't automatically power when you turn the game on ... you could play for extra balls on the previous game if you wanted to.

the paper coil wrap getting toasted is normal. That's true for any coil that stays on for a while.

the tape was used to prevent scraping the ink off when sliding out the glass, and may also help prevent heat damage and flaking. The glasses I've seen with tape seemed to hold up pretty well, but all them had tape that was is decent shape - maybe a little yellowed, but that just slightly affects the brightness of the light.

if the tape is falling off, is the ink underneath is great shape? If you look from the front, is the ink cracking?