1-a simple motor and a holder that either move the mouse or something near the sensor (beneat)
2-open up a mecanical mouse and attach stepper motor or unlimited servo to both the vertical and horizontal optical rotary encoder.
3-decode the signal emited by a PS2 or USB mouse and send you own data without any mouse
4-plenty of software online that can do the same directly on the PC without having to joggle with a mouse, but you need to download and run them.
i guess for a pratical joke, you could fit a minuature arduino inside a mouse and intercept the optical sensor before resending the data. you'll need a scope and i dont know if a arduino could do it fast enought without lag.