Hello numbskull,
Writing that doesn't sound very nice, I'm sure you're not one, This will not be a short reply but anyhow lets answer the simple questions first.
Fuses, it should only be a 1A not 2. It is unlikely to have cause any damage though. Secondly the fuse in the negative wire was there to protect the wiring from a fault that did occur on some fridges, which basically caused the negative towing power to be fed back through the gas pipework and then used water heaters negative wire, which could not handle the 8A the fridge used. If your wire has not melted, then you should be alright without the negative fuse.
Now the problem with the actual heater.
Whilst I have some pretty clear ideas what might cause the symptoms, without being able to see and probe the necessary bits its simply not possible to give you a copper bottomed solution.
I have to point out the heater is between 26 and 30 years old (only manufactured between 1980 to 1984) and they do benefit from a service from time to time.
The symptoms you describe suggest the controller either thinks there is a flame, or the water is already hot enough and does not need heating. This will give the constant green light with no apparent activity.
It is unlikely the controller is seeing a false flame signal, as it's looking for a small negative and weak voltage on the yellow wire. This is unlikely to occur without an actual flame being present, so it is not very likely.
More likely is the possibility it thinks the water is hot enough already. The water temperature is sensed by a negative temperature coefficient Thermister. This has the property that when it temperature rises it resistance falls. Set up in comparator circuit as the resistance falls below a predetermined point the circuit turns off the gas valve and ignitor circuits, and waits for the resistance to fall. If the probe has gone short circuit the heater will not turn on.
What is left is the actual circuit board, cable loom and connectors and the possibility that some of the components have failed. The most likely would be the flame detection components, but they are only likely to be damaged if something caused an HT spark to jump to the probe.
I suggest you check out www.arksystems.biz where Gary has a lot of info, and he does have some spares and offers service for some Carver products.