Another two days, and still no joy on Windows.
I just read HERE, however, that installation of Apple Bonjour is recommended as a way to help the raspberry Pi getting an IP, when connecting it via direct ethernet cable on Windows...I don't really remember if I have already installed Bonjour or not...I will check tonight, anyway, and see if installing/reinstalling it does the trick.
I'm still open to suggestions, eh...don't be a stranger, if you know something I might have missed. ;)

@Nelbrenn Is you're router set up to give out dhcp addresses? You can check if the Pi has an ip address through the Retropie menu. It shouldn't start 169 as this is the default address it'll use if it can't get a dhcp.

@stino you'll need to use the sudo command to elevate to root privileges. Without sudo at the beginning, you won't have write access to the file. Also, you'll need to open this in an editor, so try sudo nano /etc/dnsmasq.conf which should open in the nano text editor for you to mess with. Remember to save when exiting with ctrl x

@rachuism at first sorry for delayed answer..... If you hash your password then special chars are hexed and your problems may be solved with your WIFI PW.

You can read this thread and follow instructions.
Here the PW is automatically hashed.

You see in the example the passphrease "Secret Key" was hashed to "5bee4223648102a29c09b2b52bbd2fc920c4d90ced9292781477d1bf219b753a"

You can also use this side to generate hashed values..... Be careful! You give your SSID and your PW to a xenos. I tested with the values provided in the thread linked above and the output from wireshark resulted also "5bee4223648102a29c09b2b52bbd2fc920c4d90ced9292781477d1bf219b753a" .... so this works so far!

Knowledge is power!

Yeah, I'll probably go with the complete reinstall. I think I power the Pi correctly. (powering up by connecting the Pi to the Power Supply and shutting it down directly from the Emulation Station) Thanks for the help anyway! At least it gave me the clue for what might have happened.

EDIT: Yes, the reinstall did its job. I connected to the internet with no problems! I wonder what was the cause... Thanks a lot!