Thank you Dez and Medved! I'll have a look tonight (I'll unscrew the ballast from its mount and have a look that way.)
The ballast looks to be in perfect shape honestly. Coils look pristine and there is no corrosion anywhere. Also, upon disconnecting the ignitor hot/neutral leads, what I do with those wires? Do I just put electrical tape on them for insulation or do I reconnect them to another place on the circuit? It seems to me that the wires going out of the fitting and through the wall go to the ignitor first, and then only later the ballast. Though I will have to confirm this when I disassemble it a tad later. Thank you again!!
Ignitor has 3 leads: One goes to Neutral (internally connected to a resistor inseries with the HF choke), one to the lamp (the one with the pulse capacitor) and one to the tap on the ballast winding (the one with the sidac). There is nothing on the ignitor that should go anywhere else.
In a fixture could be two other components though: A power factor correction capacitor connected parallel to the mains input and sometimes the photocell (one lead to Neutral, one to the input power and one towards the ballast Line input terminal). For a basic ignitor functionality the Neutral wire needs to be on its place, the other two (the Lamp and Tap wires) could be swapoed and the ignitor will still work, just a bit less reliably (a bit lower peak voltage, less favorable pulse polarity, but still should be able to ignite a good lamp at least when cold)
From your description it looks like someone had messed up the wiring...
The capacitor has no effect on the lamp itself, its function is only to reduce the reactive power the lamp circuit imposes to the mains wiring (reactive power does not transfer any real power, it is just a chunk of energy bouncing between the ballast coil and the mains and back, just unnecessarily straining the distribution network in the process; but a single fixture is of no problem, real problem would be many uncompensated fixtures doing that together). So you may leave it out, I would even recommend that until you make the fixture working.
And regarding the meter and ignitor: Indeed, the HV pulses are extremely dangerous for a meter. And even no fuse may protect against it: There are no high currents that may trip the fuse, just high voltages causing a dielectric breakdown. And moreover the HV pulses could be the thing that may also keep an already blown fuse still conductive (by igniting an arc in it), so even when some excessive currents would be involved, the ignitor action would prevent the fuse doing its job.