In Control Panel - Network and Sharing Center - Change Adapter Settings, open the properties of your Intel NIC. Click Configure. Click on the Power Management tab. Uncheck the option that says "Allow this device to wake the computer". Give this a try and see if your system still powers on after shutdown.

It just started doing this. I changed no setting in Windows or the BIOS.
 
I pulled the system from where it was (put in some shelving units). The test bench is now on my kitchen table where it's more accessible. I'll start by removing and reseating everything (video cards, memory and CPU). Then I'll double check my connections from the PSU.

Moved the computer back to where it was (next to the desk). Played some Homeworld Remastered and Deus Ex: Human revolution. No issues while playing. I then left the computer alone. While watching TV, the computer's fans came on and the computer woke from sleep. The power settings are set to default, "Balanced".
 
It did this two times while watching TV. The displays will shut off, then the computer will go to sleep, a little while later it will turn on. It also did it this morning. I was walking to the desk after grabbing a cup of coffee (the computer was in sleep mode) and the computer turned on before I sat down.
 
I'm going to try a couple things before posting again. It may be wake on LAN but how come this issue just started when WOL had been on the whole time since the computer was built.
 
I hope another BIOS release is in the works so I can try it.

Using powercfg -lastwake in CMD, I found out it's the i217 LAN port that is waking the computer. It might also be the cause of the computer restarting immediately after shutting off.
Shouldn't the LAN port receive a signal from somewhere to wake and not just turn the computer on?

The 217 supports full WOL as well as some other things, the 210 is very similar, the same in performance, just doesn't have the extra bells and whistles, it sounds like there was some signal coming to the port even if it was being misinterpreted that was kicking it back out of sleep.  Rather odd, glad you were able to resolve it.

It's doing it again. I thought it was fixed using the other LAN port. I'll try to figure it out this weekend.
This is what my i210 LAN port power options look like. What's "magic packet" and "pattern match"?
 

 
These are the settings for the i217 port:
 

 
 
 

Good to hear

I'm using the i217 port after reseating everything and the BIOS update. It seems to working fine. I'm going to give it a few days to see what happens.

I checked eventvwr and it shows the i217 port waking the computer every 30 or so minutes. All of the items shown are related to i217.
 

 
 
 
 
 
 

My computer is set to sleep every 30 mins. It looks like it does then the i217 port wakes it back up. I haven't checked advanced power settings yet. What should I be looking for?
 

 

I just started experimenting with Sleep. Even when I turn my PC off it will restart if I just bump my keyboard unless I have wake from USB disabled in BIOS. I just manually choose Sleep instead of Shutdown if I get tired of paying for the electricity.
 

 
 

Did you disable 'Allow Wake Timers' in the advanced power settings?

Check these settings in Advanced Power plan. If 'Power Button' action is already Shutdown then change it to 'Do Nothing'
and hit Apply. Than change it back to 'Sutdown and hit Apply.
 
 

I disabled WOL in the BIOS for the i217 port and it still wakes up right after going to sleep. It no longer turns on right after clicking start, then shut down.
I didn't pull the network cable. I'll try that later tonight.

Forgot to pull the network cable last night. Woke up this morning and the computer was in sleep mode as it should be. Hit the space key and the computer woke normally.
 
This is what my card properties looks like. Should I have everything unchecked?