I’m sorry to say, it sounds like your logic board is bad to me. The Apple store can check it in and run a couple stress test diagnostics on it to confirm. But you’ve gathered enough information from a couple angles, I feel pretty confident in my analysis.
It sounds like the connector on the logic board was damaged. It happens a lot to new technicians. You can look at the connect and see what the damage is. You may be able to find someone that's good a soldering electronics and solder the connector back on for you. Otherwise, you'll need a new logic board. Update (07/22/2017): There’s not any fuses. Sometimes damage to components isn’t visible or difficult to see. I still think you have a bad LVDS connector (not the cable) on your logic board.
It's always tough to say with stuff like that. I've seen it be the cable, I've seen it be the power supply, and I've seen it be the logic board. However, if he's already had a tech look at it, then it's more likely the cable like they say. It's been about 6 months since I've had to replace one of those cables, so my memory is a little foggy, but I do remember hating doing them (which is why the tech charges $400 to replace them). I think it's just annoying because the cables run under other parts, so you end up taking out a decent amount of parts just to get the cables to run to the proper places inside the machine. It looks like someone published Apple's take apart guide for the 24"....it's pretty similar if you want to take a look: [Documento: led_cinema_display_24.pdf]
Make sure it's only shutting down when on battery power and not when it's plugged into a MagSafe adapter. Try reseating the battery connector. If it's still having issues when on battery power, then it sounds like something's wrong with the battery cells and they're unreliable. It doesn't happen often with new batteries but it does happen, I've seen then happen even with Apple's own batteries. I'd suggest asking for an exchange at that point.
Strange....maybe you have some ghosts that don't like the vacuum? A couple troubleshooting steps that might help isolate the issue:: - are you certain your UPS is working fine? May try unplugging it to first make sure it's able to sustain the devices. Then you could try running the vacuum then the machines are on battery only. This would at least help you determine if the issue is related to the power. Now for the "shot in the dark" ideas:: - If it's still doing it, I'm taking a shot in the dark by guessing there's maybe some shielding problem somewhere. May see if you can borrow someone else's vacuum to see if it does it. Does the vacuum wire every cross or get close to any of the KVM wires? - If it does it with a loaner vacuum, and it does it when the KVM is on battery backup....I'm guessing it's some shielding issue with the KVM and won't really be resolved unless you get a KVM.
If there's no bootable volume and the machine is otherwise functioning fine....you should get a folder with a question mark. In some newer Macs, I've seen them occasionally try to boot to an internet recovery partition which looks more like a globe. I hope that helps. There's a guide I put together of the startup layers of a Mac, which is for troubleshooting startup issues. Feel free to check it out here: https://www.macmasters.net/3-layers-of-mac-startup/ Feel free to contact me if you need more help.
Check your antenna cables. They're the thin black cables coming out of the display hinge. Make sure they're connected and check for damage, maybe even reseat them. The metal connectors can be easily squished, so be gentle with them. Update (07/19/2017): Assuming the connectors on the display and the connectors on the airport card look I damaged.....yes, it sounds like a bad airport card. To double check that you’re having a hardware issue....you should boot to the recovery partition (Command+R when starting up)....then in the recovery partition, you should have access to the airport/wifi settings to see if you can connect to a wifi network. You can also use a safari browser in the help menu too. Doing this rules out any software corruption that is possible in your OS.