Electrical interruption and RPM gauge going haywire

I would guess it has nothing to do with the engine.  It sounds like a clasical electrical problem. There is probably something high resistance in the positive voltage path: cracked wire, loose connection, or corrosion.   I would probably start by turing on the avionics.  Take a voltage reading at the battery, and another at something like a ciggarette ligher socket.  If there is a big difference in the voltage, look between the two measurement points for the problem by following the power wire.  Remember to turn off the avionics, so you don't drain your battery.

The last aircraft I worked on would barely turn over when starting.   The owner was  going to buy a new battery, even though the battery was only a few months old.  Anyway, this aircraft didn't use a relay, but ran the power through the main switch.  The switch was bad.  New switch and all the starting and radio problems went away.  

If it has nothing to do with the engine then why would the RPM indicator go bad if it is not electrically connected to anything else other than a single coil in the generator housing?

Whatever this is I don't think it's just low voltage or the voltage meter would be really low.

Some tachos require a 12V supply as well as the signal from the inductive engine pick-up so it definitely sounds like you have a supply or ground problem. The fact that your transponder that draws high current pulses seems the worse affected unit tells me that you probably have a high resistance connection somewhere. If you are not drawing much current then your volt meter may well read normally, it also depends on where it is connected in the electrical system.

If it were me I would disconnect the battery, then connect a multimeter set to resistance between the -ve battery connection and the -ve terminal on the tacho. I would then obseve the reading and start pushing and pulling on the wiring harness and connections. The reading should not really be gerater than 1 ohm. Don't forget to subtract the resistance of the test leads by shorting them together.

I would then repeat the above experiment but conenct to the +ve battery connection and the +ve terminal on the tacho. You will need to switch on the master switch to make the circuit. If your aircarft has a relief relay for the main supply then you will have to bridge the main contacts with a piece of wire to complete this test. Give the fuses a wiggle and the bus bar on the back of the fuse box.

This should allow you to isolate the problem and fix it. I have attached the wiring diagram from my Eurostar so you can see a typical electrical system.

Good luck!

When I turned on the beacons or landing light the voltmeter would slowly go down to 10 or 11 volts. I think you may be right!

A loose connection would also explain why the tach was twitchy right? Could it be picking up the "bounce" of the faulty connection and thinking that it indicates a crankshaft rotation?

The standard Rotax tacho uses a frequency to voltage converter circuit to drive the guage. Any large fluctuations in the 12V supply to the guage will not be protected by the internal regulator circuit and will show up as incorrect readings.