Does Arduino Nano Support Mouse and Keyboard libraries?

If not, then what is the next smallest arduino that does?

Thanks!

Leonardo? Smallest with 32U4 processor.

CrossRoads:
Leonardo? Smallest with 32U4 processor.

CrossRoads:
Leonardo? Smallest with 32U4 processor.

I am assuming that the Nano does not support it then?

Nano has FT232 for USB/Serial, it's not reprogrammable.
(or clone with CH430, I think also not programmable).

Mayne Uno with 16U2.

Arduino Micro and Teensy 2 are probably the smallest Arduino-like (AVR 32u4) things that support the standard mouse and keyboard libraries. You might also look at Teensy-LC, which is ARM based and cheaper than a Teensy2; I would expect it to have working keyboard/mouse libraries, but they might be different than the standard ones (I haven't used it, so I'm not sure...)

Pricey but VG quality:

Since we're in that subject, any of you guy has an example of programming a Leonardo to behave as a "USB Device"

I would like, on the push of a button (3 buttons), to send the following command:

Button 1 => Copy (CTRL + C)
Button 2 => Cut (CTRL + X)
Button 3 => Paste (CTRL + V)