I just bought Quake here from GOG and installed it and mostly the game seems fine, however my joystick(T.16000M) doesn't make the "moving" forward/backward movement when I move the joystick back and forth (Y-axis). Instead it only makes the look up and down ??? Why is that so and is there a way to chance the joystick movement on the Y-axis to control the back and forward movement in the game ???

The rest seems fine, I can assign different buttons to different things (run/jump/ look up/down ect.)
The X-axis works fine by turning left and right too...

Actually, from what I read it works as expected. The "go forward"/"go backward" usually are bind to keys and the mouse/joystick is used to aim. So you can shoot enemies over and under you with ease.

Okay but when I read the "Joystick.txt" file, I get a different picture of how it should work....

Quoted from the Joystick.txt file from the downloaded GOG Quake game...:
The joystick support with Windows 95 Quake has been significantly enhanced. Standard
joysticks, digital joysticks and new advanced controllers like the FPgaming
Assassin 3D, the Logitech WingMan Warrior and the SpaceTec IMC SpaceOrb are
all supported.

To make it work, just verify that your joystick or game controller is
selected in the Joystick control panel applet and has been calibrated and
tested, then launch Windows 95 Quake (WinQuake.exe or glquake.exe).

For standard and new digital joysticks, WinQuake will detect the joystick and
automatically configure it. For advanced controllers, you will additionally
need to run a config file (or run it from your autoexec.cfg) prior to the
device operating. This will set the advanced features for your particular
device. This config file should be obtained from your game controller com-
pany. The config files for the comman game controllers are included below.
If you don't want your joystick or game controller enabled, add '-nojoy' to
your command-line.

Standard Joystick Support
The standard joystick support has been enhanced to provide the following:
1. proportional movement (the farther you move the stick, the faster you move)
2. support for up to 32 buttons (JOY1-JOY4 and AUX5-AUX32)
3. sensitivity setting for each control (allows tuning and inverting the
control direction)
4. dead-zone setting for each control

The default joystick setting is for joystick left/right movement to control
turning and for joystick forward/backward movement to control moving forward/
backward.]The default joystick setting is for joystick left/right movement to control
turning and for joystick forward/backward movement to control moving forward/
backward. For optional strafing, add the 'sidestep' feature to one of your
buttons (via the Customize menu). For optional looking, add the 'mouse
look' feature to one of your buttons (also via the Customize menu).

Did you try to write -mlook in the console?

No I didn't...
But I have tried it now, that you mentioned it and it works, so thank you very much, and have a great weekend :)

Yet, I find difficult to believe you really want -mlook... oh, well.