Hot shoe pinouts / connections - do any of them provide power?

There are some cameras coming with mini flashes without a power source of their own (such as the Panasonic DMC-LX100): naturally they need to provide power on their extended flash ports. In a similar vein, there are powered accessories like A/D converters with XLR microphone inputs occupying an extended flash port: again this requires some power to be available on the connectors.

The problem is that each manufacturer has their own extensions (and often changing across the model line) to the center pin standard, and they are all proprietary and different. The MFT consortium has united on a standard for their respective models (but I don't think it is an open/published standard), but other than that, it's essentially every manufacturer for themselves.

Somewhat painful for flash manufacturers.

The manufacturer specific additions to the standard hotshoe are mostly for data transfer between camera and accessory.

The common flash connections, ISO 518 hotshots and PC ports, are simple switches that close the circuit between their two terminals. If there’s something connected then current can flow.

Electrically the design requires external power.

An accessory design requiring camera power is probably not sound in the sense that it is almost certainly simpler to add a power source to the accessory than to figure out how to draw it from the camera. It also makes the accessory suitable for more cameras.

If you only need a little power a photocell might be an option. If you need a lot of power, then the camera was a dubious power source anyway.

The two hotshoe designs with power supply contacts for small accessory flashes are the newer-version micro four-thirds (Panasonic/Olympus) and Fuji hotshoes. The power contact on both of these is brass-colored.

AFAIK, none of the other camera brand hotshoes have a similar contact. The non-sync contacts are primarily for electronic communication for features like TTL and HSS.