How does that CPU look? Has the data sentry battery leaked on the board? Grounding is also a major issue on these system 1 games so that could be giving you issues. Also never over fuse a game as you can cause further damage to the boards which depending on what damage is done can't be fixed due to some chips no longer made or available. You can ground the tabs on the solenoid transistors to verify the solenoids fire but this does not validate the transistor itself, you will need to test those also. I just finished a restoration of a Sinbad myself and was racking my brain only to end up finding out a CPU chip that you can't get anymore was bad, these system 1 machines can be a bear. Here is a link to a great article on the system 1. http://www.pinrepair.com/sys1/

I would go through that system 1 link step by step for each board starting with the power supply. If you are getting erratic behavior and display issues I'd first verify all voltages and then verify all the interconnect plugs are good. Unfortunately the ground is tied to all the boards through the plug cables, if you get a bad connection you will get a difference in ground potential between the boards which will do all sorts of weird things. It looks like the large cap (4700/25) may have been replaced on the power supply but it wouldn't hurt to put a larger one in its place and that is covered in that link. You can also pull the driver board out and then check all the drive transistors with a digital multi meter to make sure they are good, if you get a shorted transistor it can cause a coil to lock in when it is triggered causing fuses to blow. Double check all socketed chips on your board, the only ic that is socketed to my knowlege is the program chip in the top left corner labled (B) any others may have been replaced during a prior repair and by the looks of your cpu I would say is been worked on in the past. Step by step is the best thing you can do on these old boards and by all means do NOT leave the game on with locked coils, just pull the fuse on that coil till you work out the issue. Maybe a couple of the more knowlegable guys will chime in once you start getting measurements. I'm good with Williams boards but I am still a little green on these Gottlieb boards myself.

Edit: Pay close attention to A1-J6 on the CPU, it appears the battery leaked at one time and the same happened on my board and the acid ruined the metal pins in that plug. That plug goes to switches on the front door so if you can't initiate a credit or start a game that may be why. A1-J5 can also get damaged on the left hand side from acid so check that one as well.

Unlike the Williams boards the Gottlieb system 1 does not power or control the flippers from the MPU or Driver board, the flipper power should be routed through relay Q or T on the underside of the playfield. It's quite possible that one of the relays is faulty and stuck open or closed, I'm guessing its arced by you saying it works sometimes so check that relay and all the leaf switches. If it is closed manually then you will get pop and flipper power. I can check my Sinbad tomorrow to verify which relay is the one to look at. When you turn on the machine both relays should cycle after 5 seconds when the CPU boots, do you get that?

The power supply is easy to test for voltages, if the readings are off by more than a volt or if there is AC ripple the CPU may lock up, not boot or just go haywire which will make any sort of system troubleshooting a waste of time so absolutely start there. As is typical with system games you will have GI if the machine has power and the fuse is good and you can get pop and flipper power by defeating the Q and T relays. The difference between a functioning and dead game could be as simple as bad 5vdc or -12vdc to the MPU. Keep at it as it gets easier once you dive in head first!

Test the power supply in the head of the machine next to the CPU with the CPU disconnected and power on. You can check the transformers if you wish but rarely do those go bad. You will have to refer to the system 1 site for what pins should have what voltages.

As Snyper said, you are not booting at all so you need to start at the power supply and then work forward.

Looks like wires for a remote battery since the nicad battery has been removed from the component side of the board.