Unity settings and programs got messed up after fsck disk check

Question

While doing my normal activity on my Ubuntu 16.04 (Unity) machine I noticed that Firefox stopped playing sounds from any tab (Applications tab in Sound Settings had something unknown to me listed several times and nothing else). Also the keyboard layout switching stopped working, while the indicator in the panel worked fine.
I decided to reboot and after that saw the BusyBox prompt telling me that I had errors in my file system and needed to run fsck.
A similar thing happened to me about a year ago, but this time after running it on the specified drive and pressing y to every found error I typed reboot, logged back into my system and saw that the fonts are different from default and the theme switched back to Ambiance (from Radiance).
Nautilus changed its layout to (probably) non-patched gnome version, with nothing in the global menu. The 'Recent' folder returned despite still being turned off in the settings.
Image Viewer and Evince now has both global menu and local, under the title bar menu. Firefox got it's icon changed to a folder.
Some programs wouldn't start at all, including gedit and gnome-terminal. Running them in xterm outputs:

Failed to import the site module
Traceback (most recent call last):
  File "/usr/lib/python3.5/site.py", line 78, in <module>
    import os
  File "/usr/lib/python3.5/os.py", line 27, in <module>
    import stat as st
ImportError: No module named 'stat'

Other programs work fine, but some like LibreOffice or CCSM have a '90s' look instead of the correct theme.
Each time after logging in the mouse buttons wouldn't respect the settings (mine are set to RMB as primary), the settings page is correct, but after a while the screen goes black for a second and it is back to normal. Sometimes after such a 'blink' some parts the theme and fonts would change back as well.

What may have caused this and how can I fix it? Thank you.

Sorry if something isn't clear, I'm writing this late.

Additional, but possibly unrelated information:
Some strange bug causes my computer to turn on right after the shut-down. I dual-boot with Win7 and this only happens with Ubuntu, and only for the past 2 or so years. After a cursory search I was not able to find a solution to this and got used to turning it off from the back of the case right after it starts again.
During the last several months I had 3 times when my computer suddenly rebooted while I was doing something. I hadn't yet investigated that.
The disk with Ubuntu is just ~ 45 GB and currently I work with mostly less than 400 mb on the '/home' partition and 1.2 GB at '/'.
The hardware is also mostly more than 10 years old.