Why does Windows experience problems without files being obviously changed?

Sometimes the drive fails to boot at all. It would not backup, nor clone on Acronis. It would not copy using disk2VHD.exe. The drive has failed or is in the process of failing. check the event viewer for disk events. There should not be any..  If you have them replace the drive

You could from another computer run the disk manufacturers disk diagnostics. Hopefully it is still in warranty.

The files may not change, but the settings and registry do.  There are many programs interacting with each other and many updates going on in the background.  The disk could be failing, there could be memory issues, a huge number of possibilities. In some respects, it's amazing computers work at all when you think about how complex they are and how much code is in the software to make each program run.  I'm not a programmer, but I've taken some courses in Assembly language and other programming languages and dabbled on occasion.  They are incredibly complex things.  

If you're having problems it could be corrupt registry, corrupt drivers, corrupt Windows, corrupt hardware, heck, even solar flares can flip a bit and cause a crash!

Oh. So is there a section in the registry that will log a date modified change? One that I can observe?

no as the registry is always a moving target i.e. most recently used file lists

Most of the registry files are stored in %windir%\system32\config with the exception of the HKCU which is the user's NTUSER.DAT file in their user profile folder.