Think I now know part of what's going on.
Windows 8, 8.1, and 10 offer Sleep, Hibernate, Shut Down, and Restart power options. But, actually, Shut Down isn't what you might think. It isn't a complete shut down.
When you click Shut Down, or configure the Power button to power off your Surface Pro, it actually does something more like a "Sleep". Instead of writing important items to disk, and then powering off the memory -- so everything in memory is deleted and has to be reloaded from the disk at next start up -- it actually continues to power a small part of its memory, so your next restart is faster.
I discovered this because my WiFi wasn't connecting on restarting my computer. It wasn't doing a proper (full shutdown) so the WiFi had old session data, and I couldn't connect to the Internet. (You know when you've done a full shutdown, because you hear that Windows tune playing at startup.)
To fully power down the Surface Pro, you do the usual selection of Shut Down. Then, after it looks like you've shut down, simultaneously press the Power + Volume-Down buttons for 15 seconds. After a while, the white "Surface" logo will appear, then disappear, from the darkened screen. Keep holding those buttons pressed for the full 15 seconds. You've now successfully completely powered off your Surface Pro. It's no longer in a halfway house between shutting down and sleeping. So, next time you power on, it should do a complete clean reboot, and you'll hear the Windows tune.
This procedure is known as the "two button shut down". And, as it properly kills off all processes, it should prevent those strange random reboots we've been seeing.
Fingers crossed,
Rick