USB mouse/keyboard passthrough feasible?
To connect an external USB device to AVR you need USB Host supported on AVR side. Overhelming majority of AVR chips supports USB device only. There is e.g. AT90USB1287 chip exception. It even exists an AVR USB Key 2 device which can be used instead of normal Arduino unless you want make own PCBs.

Arduino UNO Mouse HID emulates USB mouse internally in Arduino. Implementing is easy when you have USB device AVR chip using LUFA library. But it's not case using external mouse. On the other side you can use legacy serial mouse or PS/2 keyboard as workaround.