Definitely get the septic tank and the pump tank pumped out before you do anything. The septic should be pumped out every three years to avoid damage to downstream components. This will also make it easier for you to service the pump.
It sounds like the system can flow by gravity if there is a problem with the pump- otherwise you would have water all over the ground. If the system was designed to be pressure dosed, gravity flow won't work in the long run. The likely reason for the pump is to completely fill the nitrification field so the water is distributed evenly- otherwise the far end won't see any water. Your local environmental health department should have a record of the installation, or if not can give you a fairly good idea.
It sounds like the pump may be locked-up, possibly due to a long period of non-use, but it could be just a bad terminal like you found on the alarm.
Normally these units have three float switches, all line voltage operation: #1 turns the pump on (high water), #2 turns it off (low water) and #3 turns on the alarm (above the high water). Sometimes a float gets tangled and gets stuck on or off. Try pulling them out individually then setting them back in.
Since the alarm went off when the tank was full you have power to the panel and the alarm float switch works.
I'm curious as to why there is a male electric plug in the panel. Perhaps it is for temporary power if the main circuit isn't working. As far as your tests with the circuit it sounds like you connected the temporary, which is probably live, to the live house circuit. That plug should be removed from the panel or isolated somehow.
I suggest that you test the pump on float and the pump itself individually. The control panel should have a "pump on" manual switch to do this.