Specifying parts for Induction Heater; how to determine current requirements at transformer?
Am I right that I don't need a high current rating at the transformer?


You need to pass enough energy to power the device, but other than that no. Induction heaters use a resonant circuit meaning the current (and voltage) in the work coil will be higher than in the supply.


  Are my hopes realistic here (financially)?


Not enough info.


  What's the danger of applying too high voltage to the work coil?


Too much current might flow, that could break something.

I'm not certain that Colpitt's is the best oscillator to use. It seems to me that this would put the resonant current through the supply filter capacitor.