OK.
Yes, it would appear that all three black boxes are relays. The outermost little relay I believe is the starter relay.
For starters, I think it's best to point out a few issues.
1. This device is programmable. What that implies is the use of a fairly sophisticated digital microcontroller that would be extremely difficult to diagnose and, if need be, replace (uc not socketed).
2. Multiple inputs from multiple vehicle operation sources, any one of which may affect the device's operation.
3. Evidence of a fairly obvious device error (uncontrolled starter operation).
4. I was unable to find a schematic for this device, without which a diagnosis of the circuits is virtually impossible.
As a point of interest, I found a New replacement unit here:
**broken link removed**
As you can see, the price is only $49.95 USD for a complete, new replacement.
While I understand your desire to repair this device, my experience tells me to suggest a simple replacement.
Now, since the only command that should actrivate the starter relay is from one of the remotes, you might try removing their batteries and replacing them. It's a long shot but worth trying.
If, however, that does not clear the error (uncontrolled starter operation), I would just replace the hardware (controller and remotes).