It took me hours but I finally got everything working including Force Feedback and decided to document it for the afterworld.
Pictures:
https://imgur.com/a/uyvcstG

The Thrustmaster F1 Force Feedback (F1FFB) racing wheel came out more than 15 years ago when Windows Vista wasn't even released yet. Surprisingly it is possible to get it working in Windows 10 (even 64bit). To increase the chance of someone stumbling on this post, the product ID in task manager is VID_044F&PID_B603 and on the package it is 2960625.

Download these files:
Official sources:

FFBRW_Vista_2.7.30.exe
2009_FFD_2.exe

Mirror:
(see comments)

Steps:

1. Plug in the device and wait until Windows says it's ready. In the Game controller settings it will only show a joystick with 20 buttons and no axes with nothing working.

2. FFBRW_Vista_2.7.30.exe right click -> Properties -> Compatibility choose Windows Vista and run as Administrator

3. Run FFBRW_Vista_2.7.30.exe and proceed with the installation until the message Plug in device comes up. Unplug your wheel and plug it in again, the message won't disappear. Wait 1 minute anyway.

4. Now go to Task Manager and !IMPORTANT! force close the Setup executable. If you don't do this it won't work.

5. Restart your computer, now under Devices and Printers -> right click Thrustmaster Force Feedback Wheel -> Game controller settings -> Properties you'll see that it is now recognized as a wheel with proper axes and all buttons/axes working. However Force Feedback isn't working and the Thrustmaster Control Panel is missing.

6. 2009_FFD_2.exe right click -> Properties -> Compatibility choose Windows 7 and run as Administrator

7. Run 2009_FFD_2.exe (keep the wheel plugged in the whole time)

8. Now - even without a restart (do one anyway) - the Thrustmaster Control Panel should show up under the Game controller settings -> Properties. Force Feedback now works as well.

General info:
Even after the installation the wheel still shows up as a HID compliant Game controller and USB Input device in Device manager. The driver in the properties will still be the default one from Microsoft. So to test if it the wheel works properly don't look in device manager instead look in the Game controller settings.