I finished up going through your original alarm/immo module yesterday. It actually cleaned up very well.
Here is a "before" picture after I had removed the relays and connector:
Here it is after getting cleaned and going through the ultrasonic cleaner with my special mix of cleaners:
You really MUST remove the relays to properly clean these and to see where damage is. Yours actually was not bad. It just needed a couple of repairs. The relays were OK.
It would be a good idea to clean up the connectors in the car, especially the black one, as those terminals had a lot of corrosion on them. I can see how that could be an issue. I would also remove and clean the ground connector under the seat, and go through the test procedure for the switch signal referenced here often (maybe particlewave posted it)? I use a PST-2 when I need to check a car, so I generally don't need to manually check switch unless I see a discrepancy between the diagnostics and what is actually going on.