Run 8 gauge unless you plan on upgrading in the future. If you do plan on upgrading, 4 or 1/0.
Are your wires colored (lol, colored)? The red one should be the power and the black the ground. If not, you'll have to get a multimeter or test light and see which has a current and which does not.
Yes, the wire with the fuse is the power wire.
The other wire of the same size is the ground, this is connected to the chassis or frame.
And finally you have a small wire, which is normally blue; this is the remote wire. When your head unit is on, it tells your amp to turn on.