If your system is actually shutting down Windows vs. a sudden reboot with no shutodwn, I suspect the processor is overheating. Try installing the Intel Desktop Utilities and monitor the processor temperature. Perhaps the heatsink isn't fully contacting the CPU? OTOH, if your system is all of a sudden rebooting with no shutdown, suspect a memory problem.
As for the LAN problem, try replacing the ethernet cable. Sometimes you just get a bad or marginal cable that sometimes works sometimes doesn't.
The shut down goes like this. Screens goes blank, system fans stop (I'm guessing system has no power) and comes back on. Next Windows comes out with warning that it did not shut down properly and pops selection screen safe mode, normal windows, etc. Funny thing is the keyboard and mouse are dead at that point. So I have to wait for the count down of windows. Once it's in the windows log in screen, the keyboard and mouse are working again.
As for the CPU cooling, I'm using a Corsair H100 watercooling kit. Temps are 39C idle and 50C Load. Load meaning test running Intel Burn-in. For the LAN cable, I tried switching to a new ones, problem still the same. Used the current ones on an another system works fine.
First lets take the power issue. Are you sure that you are supplying your PC with a clean surge protected power. Is it well grounded?
If you have a GOOD UPS try powering the PC from it.
Our systems are very much alike & I used to have the same problem until I solved it with a well grounded solution & a good UPS with surge protection.
Also make sure that the socket that is powering your PC doesn't have anything connected that drains power. Such as heater, hair dryer, etc...
PS I think the reason that you are unable to use your keyboard\mouse during boot is you have fast boot on which "in your case" disables the USB ports until windows load.