@HandySolo: Thank you for this solution and the link but I tried that _before_ putting this question here.
@wakeup: thanks for contacting me via my website; didn't know there was no way to contact me directly within here.
MY solution had nothing to do with WordPress at all. I just had a broken
httpd.conf: the line
AddModule mod_rewrite.c was missing.
So, now you might ask: How could this happen?
I was using a so called "server management tool" when setting up the server nearly 3yrs ago. After an update about a year ago it stopped working - but as I didn't use the SMT anymore I didn't really care about that because all websites were up and running - so completely forgot.
But the other day I needed to change a users PHP-flags, remembered the SM-tool and got it to run again... too bad it overwrote my hand-edited
httpd.conf-file every two minutes after that with the stuff in the database.
As it didn't affect the other websites on the server I didn't even notice, nor think to look in there! (MY FAULT, I know) So I found out by mere chance when looking for something completely different.
So please double-check your
httpd.conf twice for the entry
if using ANY script that might be accessing/modifying files in
HTH and yes, that was a perfect pebkac-error :)