1) What are the colored wires used for in a Belkin USB mouse?
For any USB device, the colors (black, red, green, white) correspond to the USB ground, 5V source, and the D+/D- signal lines, respectively. The extra black may be for the shielding ground or something.
Some USB mice also support a PS/2 connection, usually using an adaptor connected to the USB plug. The mouse detects that it's connected to that adaptor, and its internal circuitry switches to the PS/2 protocol
2) If one of the wires is for X-axis movement and the other one is for Y-axis movement, then it seems like I could simply swap the two wires from the USB cable to the PCB and achieve what I want .... correct? Basically, allowing the X-axis hack information be sent through the USB cable as Y-axis information. For example ... let s say the RED wire represented X-axis movement signalling and the WHITE wire represented the Y-axis movement signalling. If I simply switched positions of the RED and WHITE wires with each other, then would I achieve my goal?
It doesn't work that way.
The red and black lines are the power source, of course, and the D+/D- lines carry signals to and from the device. The signals are represented as a binary stream - you can't switch those wires and expect any kind of useful result.
Any wire switching has to be done at the other end of the cable, inside the mouse itself.