Welcome to Pinside.
Sounds like you have a few issues. You might have a problem in the games switch matrix. Also sounds like you have a problem with a solenoid blowing the playfield fuse. Do any of the solenoids look cooked? Can you pinpoint which solenoid is causing the fuse to blow?

Regarding the switches, get into switch test mode, remove the ball from the outhole and make sure all drop targets are up.
The game manual (electronic pdf page 21, paper page 17) lists all switches and their test number.
Using the manual, activate each switch in reverse order, i.e. from switch 40 down to switch 1. The reverse order is important because the switch test will only display the lowest number closed switch. Push the drop targets down to activate them and leave them down - they will play a part in the switch problems if any.
Note down any switches that report the wrong number and come back with the results.

It almost looks like the playfield and cabinet connectors are incorrectly wired at the MPU board. Is there any chance you can post clear pictures of connectors J2 and J3 at the MPU board showing wire colors? J2 goes to the playfield switch harness and connector J3 is the cabinet switch harness.

Can you confirm whether the following switches report correctly or not in switch test mode?
13... Saucer Drop Target
21... Drop Target #1
29... #4 Target
37... Left Slingshot
32... Saucer

Just for reference, below I've attached the switch matrix wiring diagram - click the picture to expand it for clarity.
Note there are two tilt switches in the cabinet and one tilt switch on the playfield near the flipper mechanisms. Also there is a slam switch on the coin door, and another slam switch near the tilt plumb-bob in the cabinet.
Numbers in Red indicate the switch test numbers.
Switches marked in Blue are cabinet switches and their wiring details. Wiring details in Green refer to playfield switches.
You might notice your lower three rows of switches in the matrix seemed to have moved up one position in switch test mode.

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RollingStonesSwitchMatrix.png

Ah, ok you've got some battery corrosion on that board which is likely causing resistive shorts on the switch return signals.
Post a picture showing the whole MPU board if you can.

If you have a multimeter, please perform the following:
Disconnect J2 and J3 from the MPU board.
Set your multimeter to high resistance mode. With the machine OFF, at the J3 pin connector on the MPU board, measure the resistance across pins:
9 to 10 =
10 to 11 =
11 to 12 =
12 to 13 =
13 to 14 =
14 to 15 =
15 to 16 =

Fill in the results. You should measure around 110k ohms resistance across each of these pins at J3.

Hmm, the measurements look ok.

If you reconnect J3 but leave J2 disconnected, power on the machine and run the switch test, do you still see the errors with the Credit, Tilt and Slam switches?

Also then reconnect both J2 and J3 and power on the machine. Get the machine into switch test mode and then pull off the J3 connector from the MPU board live. Do you still see the errors with the playfield switches?

Another thing I would try is swapping the U10 and U11 chips - i.e. put the U10 chip into the U11 socket and U11 chip into the U10 socket. They are the same devices (6821). Take care to reinstall in the same direction - notch towards the left.

Now that's weird. J3 has nothing to do with the playfield switches and should not stop them working. You are getting "0" in the ball in play display when no switches are closed right?

Your picture shows U9 and U10 - they are certainly different.
You want to swap U10 and U11 - your U10 chip is a 6821, your U11 chip is a 6820 and yours are made by different manufacturers which is why they look different - they are however essentially the same functional devices.

What's the history of this machine? Did it ever work in your possession? If yes what changed between the moment it worked and then started failing?

You're still getting all the other weird behavior too right? You mentioned you changed the Solenoid Driver Board - do you have a spare MPU board to swap in?

Don't suppose you have an EPROM programmer?

If your Mata Hari has a factory "-17" MPU board, it's too messy to convert to support the Rolling Stones ROMs. If you're lucky maybe the old board was replaced with a later "-35" board. If you post a picture of the Mata Hari MPU, we can tell.

It's most likely a MPU board problem anyway so let's verify it the easier way. Put your Mata Hari MPU board as is (leave the Mata Hari ROMs on it) into the Rolling Stones machine. Before you power on, make sure all drop targets are in the up position.

Power on. Ignore the fact that lamp sequence will be wrong, you won't get sound and some incorrect solenoids might activate at game initialisation.
Go straight to switch test mode and check if those switches are still being reported incorrectly.

You can do the same by putting the Rolling Stones MPU board as is into your Mata Hari machine - again make sure all drop targets are up first. See if you get wrong switch numbers for credit, tilt, slam.

Can you post some clear pictures of the back of the Rolling Stone MPU board? Also more pics of the front of the MPU board particularly where the corrosion is - i.e. bottom left, bottom middle and bottom right.

We might try checking a few more things with this board but I've gotta head out now so will come back tomorrow on details and after seeing more pics.

The second pin up on J2 is not seated properly , I think its Brown/White pin 14 .
This wire connects to TILT, Left Outlane, Drop Target B No'2 target and Right bumper. Try pushing it back in fully and see if that changes anything,
Id double check the switch wiring too . Your board looks very corroded . that should be fixed up . And the pins could be tarnished or corroded on JI and J2 this causes all sorts of grief , last resort . Try swapping another 5101 into the board. id found faulty 5101 chips can cause all kinds of problems ,
Good luck

Surprisingly it looks like most of the corrosion is around the edge of the board and hasn't hit many components. Of course there could be stuff going on in places we can't see: under the J3 and J4 pin headers, under the U7, U8 and U11 sockets, under the dipswitch blocks, etc.

Realistically all that corrosion needs to be removed and then the copper protected. I solder over any bare copper to put back more "meat on the bones" since the corrosion eats the copper. Replace any components showing corrosion. Provided the repair is done properly and thoroughly, the board will be very reliable.

You might get some inspiration from this thread:
https://pinside.com/pinball/forum/topic/stripping-battery-corrosion-with-a-strong-acid

BTW when you replaced the U8 socket, did you properly clean up all the corrosion around the socket area on the board? Corrosive crud is conductive and must be removed.

Most likely you need to change jumper wires to configure the replacement board for your Rolling Stones ROMs.

All three of your Rolling Stones ROMs are 9316A type which is compatible with 2316 type ROM.

The list here describes which jumper wires need to be connected for different ROM types:
http://www.planetarypinball.com/mm5/Williams/tech/bally_read1st.txt

Below I've indicated where the wire jumpers are on your original board. If the replacement board has other jumper links where there are "E" points, they need to be removed.

RollingStonesMPUa.jpg

The chips at U15 and U16 are correct. U15 is a MC3459P and U16 is either a 8602 or 9602 (8602 operating temperature is between 0 and +75 degrees centigrade, 9602 is between -55 to +125 degrees centigrade).

There's a trace that's been ripped off during the repair between the pink arrows below: Compare against your original board. These two chips create a master clock signal to the CPU chip at U9. If this clock signal is non functional the MPU board is brain dead and will power up with a locked on LED. No clock signal to the CPU means it can't execute code from the ROMs. The very first bit of code normally executed is to switch the LED off.

MPU board with missing trace in clock circuit

There's probably a good chance the 8602 is fried. Is it hot to touch when you leave the MPU board on for a few minutes?
Do you have a logic probe?
Do you have a multimeter that measures frequency (Hz)?

For now measure the voltages in the clock circuit as circled below.
Bally MPU board Clock circuit with oscillating average voltages

Zero volts on both output pins (7 and 10) of U16 means that chip is not oscillating. Looks like it's dead.
The voltages on output pins 8 and 11 of U15 look bad too.
For now closely inspect the previous repair work looking for accidental solder shorts and damaged traces not effectively repaired. You can try hooking up your logic probe to confirm if those two output pins (7 and 10) of U16 are pulsing or not.

What's the "2.2" written next the +5 in the middle of that area about?

May as well grab another 9602 and MC3459 (alternate replacement) chips from Great Plains:

https://www.greatplainselectronics.com/proddetail.asp?prod=9602
https://www.greatplainselectronics.com/search.asp?pg=1&stext=MC3459P

Pin 14 of U15 measured 2.2 volts? It should be 5 volts since it's connected to the +5V power rail.

Yep, grab the 74S37 instead. Grab some replacement IC sockets for both those chips. The one installed at U16 looks like an old low quality socket. But don't replace the sockets yet - leave that till last so you don't introduce new problems. Get the board working first with the replacement chips.

Connect the black alligator clip of the probe to ground (TP4 (top right) on the MPU board is a ground point)
Connect the red alligator clip of the probe to +5V (TP5 (bottom middle) on the MPU board is the logic +5V power point)
Set the switches on the probe to TTL and away from Latch.
Power on the MPU board, put the tip of the logic probe on each of the output pins (7 and 10) one at a time of U16. You're looking for the yellow Pulse LED to be on at each output pin.
You can then try probing the same pins on your Mata Hari MPU board to see how the logic probe should react.

You might find this useful:

https://pinside.com/pinball/forum/topic/terrybs-guide-to-logic-probes

Voltages on U15 look better now.
Make sure connections to resistor R4, R10 and capacitors C14 and C15 are good. They are critical to the 9602 clock circuit running.

Yeah pretty much. Replace the 9602 first and if no go then replace the 74S37 too.

Good to hear.

Any chance you can post a video of the lamp/sound issue?
Lamps are handled by U10 on the MPU board, sounds by U11 so there's no shared signals.
Are you able to tell if it's specific lamps being activated that are causing the sounds?

What did you touch/change?

You can't buy new 9316 chips. You might find someone on ebay selling old Rolling Stones 9316 ROMs.
Todays option is to get 2716/2732 EPROM replacements that have been programmed with the game of your choice. There are a few sources (ebay and probably some folk here). You will however need to change jumper settings on the MPU board to support these EPROMs, because they have slightly different pin configuration compared to 9316 ROMs.
Alternatively if you're keen, you might even consider buying an EPROM programmer and EPROM chips to do it yourself. That way you can program the EPROMs with whatever game you want at any time.

If you look closely, all three of your ROMs (U1, U2 and U6) are 9316A ROMs with Ballys part number stamped on them.
E-796-17 U1 (Rolling Stones game code part 1)
E-796-18 U2 (Rolling Stones game code part 2)
E-720-35 U6 (Bally version 35 system code)

The data in 9316 ROMs is implemented as part of the manufacturing process of the chip. They cannot be reprogrammed.