Swarlos

Also, for clearing CMOS, I don't know why everyone says to pull out the battery.  Most motherboard have easier ways to clear it.  Including yours.
 
Picture taken from ROG_STRIX_X470-F_GAMING Manual
 

Have you tried using different USB ports for the keyboard? Do you have an old PS/2 keyboard somewhere you might be able to use?  I'd be surprised if the CPU or RAM is the problem.  I don't think you'd get it to POST at all if either of those were the issue.
 
Edit: Also, the Q-LED in the top right is Green so I think CPU and RAM are fine.

What Lurick said, there's probably Ethernet drivers that you need to install on a disc that came with your motherboard.  You can probably download them off the Gigabyte website on another computer if you can't find the disc or if you don't have an optical drive.

Have you tried using different USB ports for the keyboard? Do you have an old PS/2 keyboard somewhere you might be able to use?  I'd be surprised if the CPU or RAM is the problem.  I don't think you'd get it to POST at all if either of those were the issue.
 
Edit: Also, the Q-LED in the top right is Green so I think CPU and RAM are fine.

Did you install the ethernet drivers for the motherboard?

Sometimes you need to push the board toward the back a little bit to get the screws to line up.  Check to make sure the I/O shield isn't getting caught on the case opening in the back.  Just make sure you don't scratch anything.

Have you by mistake, switched to another display in the built in windows option.
How about this, start the computer wait for like 1 minute and then press f1 once, then press the down key once and then press enter once

Do not put pcie cable into the 8pin CPU power connector.  How old is that PSU?  Usually a PSU comes with a CPU power cable with two 4pin connectors that you put in together to fill the 8pin connector.

Those are good temperatures.  Idle temperature doesn't mean much of anything as long as your load temperatures are good and low, and 53c under load is quite low.  If you want to bring idle temperature down you can set the fan curve on your CPU to run faster at lower temperatures.  I think your CPU fan is just running really slow at idle because it doesn't need to go any higher.

Those are good temperatures.  Idle temperature doesn't mean much of anything as long as your load temperatures are good and low, and 53c under load is quite low.  If you want to bring idle temperature down you can set the fan curve on your CPU to run faster at lower temperatures.  I think your CPU fan is just running really slow at idle because it doesn't need to go any higher.

Also found something about the green light in the manual.  It's a +5V Standby Power Indicator LED.  It basically just shows that power is being supplied to the motherboard.
 
 

I googled the manual for DQ67SW motherboard.  I'm seeing that it officially supports RAM modules up to 4GB.  I don't know if your 8GB of RAM is two sticks of 4 or one stick of 8GB.  I don't know if the manual is outdated, but that's a potential problem.  
 
Also, I'm seeing that motherboard officially supports the 2000 series CPU's, the Sandy Bridge 1155 chips.  That could also be outdated, but another potential problem if the board doesn't support the Ivy Bridge 1155 CPU you have.

Wild guess. The motherboard what you have doens't have an up-to-date bios which result in the system  doesn't boot.
 
A320 is older an chipset. It will only work with R 2xxx if you have an up-to-date bios

Can you get into BIOS or do you not get any video at all?  Does your motherboard have a display for troubleshooting codes?

Can you get into BIOS or do you not get any video at all?  Does your motherboard have a display for troubleshooting codes?

I'd recommend carefully reading the instructions that came with it, but I found a video that might help you.  I don't know if that's the same CPU that you got, but it should be the same heatsink with the same installation method.
 
Edit: If you mean the bracket on the backside of the motherboard, yes that stays on there.  If you mean the two plastic pieces on the frontside of the motherboard, I think those are used with the Wraith Prism heatsink so you remove those.
 
 

The CMOS battery holds the motherboard configuration settings, following the steps Master Disaster said will clear the configuration settings and the motherboard will return to default settings.  There's probably a clear CMOS jumper on your motherboard too, but removing the battery works.

Extra sticks does increase the loading on the memory controller. Maybe this pushes it past the limit in your case. Assuming you are/were running 1866, you can try backing off to 1600 with all the sticks in place. The other option is to fiddle with voltages but I'm not that familiar with DDR3. If it was DDR4 I'd look to bump up VCCIO and VCCSA. It helps for testing purposes if you can find a way to make the system crash quickly so you can check of changes made a difference. Something like either aida64 mem only stress test, prime95 with lots of ram selected, or memtest.

The CMOS battery holds the motherboard configuration settings, following the steps Master Disaster said will clear the configuration settings and the motherboard will return to default settings.  There's probably a clear CMOS jumper on your motherboard too, but removing the battery works.

1) Unplug the power from the wall
2) Remove the button cell battery from the board
3) Leave it out for a few minutes
4) Replace battery
5) Reconnect power
6) Boot

I don't see it specifically noted on this page on the Intel site. Xeon isn't listed in the Ivy Bridge section, it starts with Haswell.  Being able to handle a large number of cores doesn't mean it supports your specific CPU.  I don't know if it's definitely not supported, but I can't confirm that it is.
 
https://www.intel.com/content/www/us/en/support/articles/000006105/processors.html

That sounds like a pretty cheap price.  Whatever cooler you get make sure it's compatible with the 2011 socket.

I7 3770 or 2700 are easy to find on eBay, delivering similar performance.

/agree with W-L
 
If all that was removed from the stock cooler thermal compound was that little smudge on your hand I wouldn't worry about it.  There should plenty left on there to get the job done.

Probably won't be affected at all unless it was completely smeared off.