I'm assuming here that you're powering the chip from the PICkit rather than from an external circuit? I... have had weird results trying to power circuits from my ICD3, the driver invariably crashes for some reason. For that reason, I always program in an externally powered circuit. I don't know if the PICkit is any better or worse, but it wouldn't surprise me if that route is a bit troublesome. In any case, the chip should definitely be able to program in a circuit with a 5V supply voltage.
This is just from instinct, not expertise, but the problems you're describing sound fiddly enough to be power problems, so I'd reckon there's reason to hope that a bigger regulator will yield better results.
I'm glad you've had some success, though! I'm wondering if the behavior you're seeing with only the one partition showing up has something to do with your using System 7... my data points are limited, but I've so far tested this on all three of my machines that have floppy ports, and both the 512ke and Classic (both running 6.0.8 or earlier) show all the partitions, while the Classic II (running 7.x) only shows the first. This could be because of the system version, the ROM, or the processor speed, but I did notice that in System 7, the emulated HD20 had a floppy disk icon in file dialog boxes, which makes me suspicious of how it's handled by the operating system. In any case, it's worth testing to see if you get all the partitions if you boot into 6.0.8 instead of 7.x...