While I've come across a Windows.old folder after a repair install, I've never seen those .00/.01 folders.
From a Google I found this answer from a MS tech on another forum, so it looks like you can delete those.
"The found.00% folders contain recovered clusters by Scandisk or Chkdsk in the event of a corruption in the filesystem, you can view their contents if there is any data that is still recoverable, otherwise if access permits it, you may delete them"
Did you try the cold boot to see if it resolved the F2 or F12 problem ?
I wonder if turning off fast start up may allow you to get into F2 and F12.
You haven't said if you are using a laptop or a desktop machine.
Go Start - type powercfg.cpl and press enter when it comes up.
With a desktop machine you would click on Choose what the power buttons do but on a laptop it's Choose what closing the lid does.
Click on Change settings that are currently unavailable which will activate the boxes below for fast startup and hibernate to uncheck.
Click on Save and then shutdown and try F2 or F12 again.
As you reinstall each program, run another sfc /scannow to see if any produce those errors that sfc /scannow is popping up with.
Which antivirus program are you using because any 3rd party AV program should disable Windows Defender.
I've recently changed from Norton Security to Kaspersky Total Security and subsequent sfc /scannows have come back clean except for on one machine.