somuchfun

Update:
10 days or so ago (April), my WordPress (4.4.2) magically fixed itself and all of the above mentioned issues disappeared.
I can’t explain how that happened, since after those problems occurred and after my failed attempts to fix them back in February, I’ve basically left it be.

After this joyful discovery, I was of course wary of upgrading to the newest version 4.5, but finally did, and everything appears to be in order.

…and now the Appearance stopped working: wp-admin/themes.php gives the same error:

Internal Server Error

The server encountered an internal error or misconfiguration and was unable to complete your request.

Please contact the server administrator, [no address given] and inform them of the time the error occurred, and anything you might have done that may have caused the error.

More information about this error may be available in the server error log.

My website is working, but so what? No updates or changes possible. I have no idea how did that happened. Something must have gone wrong on Dec 20th (bad 4.4 update?) and I don’t have a backup copy. I have zero knowledge about these things, so unsure what to do next.

Thank you for the suggestion, CureWP.
I renamed the plugins folder to plugins-bak and the plugins have disappeared from my website, but I still cannot access the update-core, settings (general options) and updates on my Dashboard.

Update: So I manually updated to 4.4.1, but the same problems persists (no access to plugins, update-core and options-general php).