Clean and adjust the switches on the ball release gate. There are three switches. Manually open and close the gate, observing the correct make/break of each switch in each position. Does the 110V coil for the trip bank reset energize? It is fused, so check that fuse if it isn't firing. If the fuse is good, check the "armature" switch. This switch is opened and closed by the action of the start relay's armature plate.
You can access it easiest by loosening the wing nuts on the reset bank, and flipping the bank assembly over.
It's carrying line voltage, so check it's continuity with the game unplugged from the wall outlet. These often become misadjusted or need cleaning, and the trip bank won't reset unless this switch is making.

Here are some photos of the ball gate switches and the armature switch.
Sorry about the blurriness of the first photo. Please note, these photos are not from a "Sky Line", but from a "Bowling Queen". Should be exactly the same switches in both games.

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Nope. Five balls and five balls only. There may be an adjustment for three ball play, but I've never used it on this era machine.

Replays, which are won by hitting "special", reaching a score threshhold, attaining a certain number of "advances", or by getting a "match" at the end of the game, are simply free games. Any time the credit wheel is anywhere but zero, the game can be started by pressing the button on the cashbox door. At "zero", a coin is needed to either start the game or add credits. Many of us modify the switch on the replay unit which opens at the zero position, so that it is closed at zero. This way, the game is always on "free play", and a coin is never needed to start the game. Each and every game though is five balls, because this is a replay machine. There is no add-a-ball counterpart to "Sky Line", although I wish there were. I've often thought about what it might have been called. Maybe "Sky High", "City Lights", or "High Society".

Man, this brings back memories. Between my brother and myself, this was like our fourth or fifth game ever. He bought a pretty wrecked one, and brought it back to life. The front of the head was torn up a lot worse than yours, and he made a whole new front panel for it out of 1/4" plywood. He has the game to this day.
There may be a common cause for the three issues in the head. You said you cleaned all the jones plug connectors, but I'm thinking there's one pin and socket that isn't making. I don't have a schematic in front of me, but look for something that is common to the subtract credit, reset animation unit, and the add credits circuits. I know I had the same problem of not subtracting credits on my "Bowling Queen" a while ago, but I can't remember what fixed it. It was just one switch that needed adjusting, but I don't remember which one it was.

It sounds like you may have gotten a bad batch or two of lamps. Try a different brand if you can find them. If it still continues to blow out bulbs after using a different brand, you might have some kind of intermittent leak from the solenoid voltage into the lamp circuit. Not that it would affect the lamps, but just wondering; is the game high tapped?

Have you checked the voltage of the lighting circuit with a DVOM?
You say the lights get bright. Is that a momentary thing or do they stay bright for a while before burning out?
If it's something that you can measure, you can determine if it's a current leak, or a bad batch of lamps.
Gottlieb used what are called "sand resistors" for some lighting applications. These allowed them to use a 6 volt lamp in a 24 volt circuit. I'm not sure if "Sky Line" uses any of these anywhere, but if so, it's possible that one may be breaking down and allowing 24 volts to leak into the lighting circuit.

That isn't the typical sand resistor that is normally used. You won't get a true resistance reading unless you unsolder one leg of the resistor. Looking at your schematic, it doesn't look like this resistor has anything to do with lighting anyway. It's probably there to prevent overcurrent from pitting and burning the rivets on the wiper disc of the 0-9 unit.