Basic Wiring

[yarisugi]

You would have to connect the (+) to a power source and the (-) to a ground or some metal part connected to the body of the car.

You should've bought something like this, since it just plugs into the cigarette lighter socket.

The one you have is plug-in but the Yaris doesn't have that type of socket.

To answer your questions:
1) No conversion needed.
2) 14-18 gauge wire is sufficient, size doesn't matter
3) you'd want to have an in-line fuse which you can get at any auto parts store.