Assigning buttons to look around...
You've mapped the buttons to the wrong items. Left/Right/Up/Down (by default the NumPad arrow keys) controls the mini-menu, which allows you to add friends, adjust the radio, and change the driving aids. If you go back into Options - Controls, you should find options for Look up (Head), etc, as well as left, right, down).
In the last tab, you can adjust the sensitivity and linearity of the stick. There is also a separate sensitivity slider for bikes (Bars).
Use your driver software to configure the deadzone.
Needless to say, Joysticks are not designed for driving games. I haven't used a Joystick for a driving game since the mid 90's. You are much better off getting a gamepad (best on available is the XBOX 360 gamepad for Windows) or a steering wheel (most popular wheel here is the Logitech G25) (Obviously your budget will come into play here). I would still recommend a keyboard for driving games before a joystick.
I have a Logitech 3D Extreme joystick for FSX and can't make use of the hatswitch in that game either, although I can use the POV control on the G25 to look around in TDU and other games when I'm not using TrackIR.