Yes the ground braid is just for safety, it's not required to make the game operate.
Read though the online repair guides and check it all systematically. Fuses, power supply test points should all be checked. You'll probably need at least some connector repinning, adding the various reliability mods, and replacing the +5VDC logic supply filter cap.

http://techniek.flipperwinkel.nl/ballyss/rep/index1.htm

Check to make sure there is no alkaline battery leakage (post photos if unsure). Usually connectors are a big problem, so you'll need to learn how to repin and aquire the tools if you have never done that before. Start with getting the power supply in spec first (measure with a voltmeter/DMM) make sure all the fuses are the correct value and functional. Post about any further questions you need answered.

I am unsure what you experienced there, as all the voltage sends and returns are through the harnesses. I've also had several Ballys in this series that had both the braid disconnected from the backbox and the ground prong missing from the mains cord. They ran well for many months before I found that the braid wasn't even connected.

No, the metal backplane on the backbox is not the return itself, but is technically the ground "shield". This has more to do with noise reduction. Ideally, you don't want current to flow through the shield (i.e. the metal backplane). Bolting the boards to the metal shield helps reduce noise. The short answer to the question "will the game operate without the shield?" is yes. But it *possibly* can have noise issues - the degree of how much they can affect your logic, or more commonly - the audio performance - of your game can vary. There is some in-depth information here which can better explain the grounding practices:
http://www.rane.com/note151.

I believe most slam switches are normally open, at least the ones I am familiar with. They are usually a part of the switch matrix (this is true for Stern Quicksilver, so should be the same for Lightning)

The battery area area doesn't look bad, a bit difficult for me to tell in the pics. Usually there are no diagnostic LED flashes if there is a problem with the reset circuit in the vicinity of the battery. Check the +43vdc power, if that is not present, that could be a problem.

As far as the 2nd pic, I don't see a problem there. Cab you explain your question?

Go back to my first post and click the link

MPU status LED test flashes...how many/what's it doing there?

If it isn't getting to attract mode, it won't be polling the switches, so it won't be able to read the switches for the trough or coins.

I think Dothedoo is right about the balls. Now you have me questioning the presence of balls to start attract mode.... I'm not at my game to verify.

How about coin door switches to put test mode? This would verify the MPU is at least running - then from there you could enter switch test.

I checked the relationshio of attract mode to presence of balls on my Bally Space Invaders. This is the only -35 MPU game I have right now. When I powered up with the ball out of that game it went into attract mode just fine, and found that the ball is not required to either enter attract mode or start game. That's interesting because this board set is used on a lot of games...so is it software dependent? Who has some different early Bally/Stern that can check and comment?

Usually I take a jumper over to a GND test point on one of the boards. I find the side rails inconsistent, more often under the playfield I would use the ground braid (always make certain that the ground braid has a connection up to the backbox metal plate behind the boards - as often this one gets disconnected or broken when removing backbox heads to move games)

Totally backwards from what I think it should be.

Think about this: Let's say you have those two games on location, one is a single ball (like Space Invaders) and one game is a multiball (like Centaur). The single ball game is essentially dead without a ball, so if it didn't still go intto attract then it would be easy to know it isn't going to play and requires service. If a multiball game, it could potentially play with less balls (of course, this would depend on programming it to acknowledge missing balls and adjust gameplay to accomodate less balls than normal) and thus the multiball game could potentially still earn money for the operator...even with a lost ball.