So, what am I looking at here? I removed the cap, and then a flanged nut, and I'm left with this protruding bolt which I assume is poking out of a tapered BB. The problem is that it protrudes too far for my crank puller to fit.

Do I need a different tool?

You have a square taper crank with a nut-style spindle instead of bolt-style spindle. You use a standard crank puller, but you need to back off the "pusher" part of the puller all the way. In fact, take your puller all the way apart. Then thread the outer piece into the crankarm. You may find that it's possible to thread it most of the way on and still get engagement between the "inner" and "outer" pieces of the crankarm puller. Remove normally.

If the pieces won't engage, a different puller might. I bet a Park CCP-2 will do the job: Park Tool Co. » CCP-22 : Crank Puller for Square Taper Cranks : Crank Tools
Park Tool Co. » ParkTool Blog » Crank Installation and Removal- Square Spindle Type

as mentioned my Park ccp-2 puller recesses enough, i think, to do the job. i know for certain i've got one of those old spindles in a box somewhere that i probably took off an early 80's Miyata. i HAD to have removed it, because i KNOW i never bought one.

I've removed 2 cranks like that with the Park CCP-22 it does dissemble.

The Park crank puller I have like this
Park Cotterless Crank Puller - CCP at REI.com
you just need to twirl the blue handle counterclockwise to back it out of the screw-into-the-crankarm part and you will be able to screw it into the crankarm without interference from the push-on-the-bolt part.
(well, you know what I mean)

I ALWAYS pull my puller completely apart - I have seen too many cranks with damaged threads do the someone not complete engaging the puller before cranking on it. Once apart; thread the part completely into the crank, then install the pusher part of the tool.

Thanks guys... I've got a cheapo Spin Doctor puller. I didn't want to invest in a fancy one since I only plan on using it occasionally.

Anyway, I backed it out, but my next problem is that the threading seems not to fit. The thread on the tool looks finer than in the crank. It's an old Peugeot PH8, but it looks like an upgraded crank... Well since the 80's anyway.

Is it possible I have some weirdly threaded crank? Or a weirdly threaded puller?

The crank just says "custom" on it and "Japan" on the backside.

VeloBase.com - Component: Sakae/Ringyo (SR) Custom

As ClarkinHawaii notes, you have an SR "Custom" crank, which uses a standard 22mm x 1mm extractor thread. Otherwise, a Maxy-type extractor designed for those nutted spindles was available:

Thanks guys! I think I'm going to invest in a decent puller... this one screws in about 1 thread and stops, while the dustcap screws in easily so I think the threads on this puller are wonky.

One more question.. is the left-side spindle nut reverse thread? The right side came right off.. the left is stuck tight and I want to make sure I'm moving in the right direction before I really start cranking on it.

Both sides are right-hand thread.