thermostat causing heating element to click on and off repeatedly
I initially misread the symptoms -- if the fan is indeed running properly with the system ON, then your system is cycling on its high limits.  I'd check for air obstructions (such as bad ducts or a dirty/too-tight filter) or a sluggish fan motor.  You can also temporarily jumper the high limits for testing purposes, DO NOT LEAVE THE SYSTEM UNATTENDED when you do this, and keep an ABC or BC dry chemical extinguisher handy in case it does start to smolder.

As to the fan issue, that's an issue with the NC side of the fan relay -- it can be bypassed by setting your thermostat from "gas" or "fossil" mode to "electric" mode, where the thermostat is responsible for calling for both fan and heat instead of letting the furnace call for the fan.

If the symptoms were what I thought they were, with the fan and the heat cycling together, than they'd sound like a faulty connection at the C (0V/return) end of the air handler's blower relay -- the return from both heater contactor coils goes to the blower relay return end instead of returning directly to the C (0V) end of the transformer.  When the G (fan) line is off, the connection between the grey C wire from the heat kit connector and the rest of the grey wires is not made, causing the heat to not turn on at all as the contactor does not energize.  When the G (fan) line is on, the operation of the blower relay causes the dodgy connection to make for a short period of time before it comes undone again, and this cycle repeats itself as long as the blower relay is on.

The "click" you are hearing is most likely the contactor itself, and that's normal -- the short cycling is the problem.  And yes, you can safely jump R to W and/or G on your unit -- that won't hurt it, as that's what the thermostat's contacts do anyway.  As to troubleshooting the problem, I'd try measuring the voltage between the grey wire coming from the heat kit connector on your unit and the grey wire at the 24VAC transformer.  It should be very close to 0V at all times; if it reads near 24VAC, then the connection at the blower relay isn't working.

P.S. your air handler is really a Nordyne -- a wiring diagram for a similar unit can be found here.