Help catching a hacker

First thing to do is Reset your Internet Router. No one can get to your computer unless something is installed on your computer to allow it, by logging into (Accessing) a website, or your Router is setup to allow access to that computer and there is something installed on your computer that makes changes to allow it.

Wiping the drive and doing a clean install is a start but you should also check the router for any Port Forwarding settings and delete them (unless you set them up for a specific reason) or just Reset the router back to Factry defaults and re-enter your settings.

First, note that if the computer was actually fully powered down (not just sleeping), it is impossible for a hacker to remotely turn it on, unless that hacker has somehow modified the physical hardware of your computer.

It is possible to set your computer to start up on a schedule. Unfortunately, since you wiped the hard drive, we have no way to check to see if that was done. An alternative explanation would be a hardware issue, which could also fit with the "remote control" issues. Although remote control is certainly not impossible, it's unlikely in most cases. On the other hand, I've seen many, many cases where people were convinced they were being remotely controlled and it turned out to be something much less interesting, like static electricity buildup, a swelling battery or some malfunctioning software.

For more info on the remote control thing, see:

My mouse keeps moving around on its own, as if someone is remotely controlling my Mac!

As for keeping your Mac safe, it sounds like you're already doing everything right. Your Mac, as you describe it, should not be at all susceptible to remote hacking.

The one mistake you're making is delving into the Console for clues. That is a black hole from which those having paranoid thoughts seldom escape. There's a lot of weird stuff logged in there that is perfectly normal, but that looks scary to those who don't have a real understanding of what they're looking at. Unless you're a Unix guru, you'd be wise to stay away from the Console.

As for the weird things you're seeing, the screenshot thing is something that is done internally when your machine goes to sleep. Settings have been known to get changed mysteriously for some people following system updates. No idea why, but normal. The Bluetooth thing probably represents a log of computers found nearby at some point when you had Bluetooth turned on.

As for advertising, where are you seeing that? In e-mail? If so, I get all kinds of ridiculous spam all the time that has nothing at all to do with my interests. That's normal. Keep in mind, if you're wondering how they got your address, that the Windows computers or e-mail accounts of anyone you have ever exchanged e-mail with could easily be infected/hacked, and then all the e-mail addresses could be harvested to be sold to spammers. Basically, once you have shared your address with people, it's only a matter of time before it starts getting spammed.

I've probably seen a few thousand reports similar to yours on this site. In only one case that I can recall did it turn out that there was a genuine intrusion, and that was via a hacked Hamachi account.

If the 'screenshot' messages are

device_generate_desktop_screenshot and device_generate_lock_screen_screenshot

They seem to show each time the computer sleeps. Nothing to be concerned about.