The colors are nice, but not always that useful. For a DC motor if you apply power it will spin in a certain direction. If you apply that power in reverse it will spin in the opposite direction meaning there is no "correct" way to connect it. So it depends on how you mount it and your code. If you have it mounted the way you want and don't want to fiddle with code I would just swap how the wires are connected to your motor driver.

Yes they all spin. The skinny metal shaft is typically used to add a wheel encoder. The wheel encoder allows you to determine how far you wheel has turned. This metal shaft turns at the same rate as the motor and is not effected by the gears. For this motor for every turn of the wheel (the plastic shaft) the metal shaft will turn 48 times (since its 48:1 gear box).

The other two plastic shafts are to attach a wheel. I only use one of these shafts to attach one wheel. The other plastic shaft goes unused in my robot.

These were definitely not designed for ease of use. There are two through holes where the motor mounts to the transmission. I think they are 3mm. There is also a small tab at the center front of the gearbox with a hole. You need to use bolts or sometimes I just use 3/32 pop rivets if I am not planning to remove them. I can speak from experience that the gearboxes are made from some sort of high molecular weight plastic (I'm guessing polypropylene or polyethylene) and no glue will stick to them---I've tried pretty much all the choices. VHB tape might work for temporary installation.

Yes.

You need to solder on the wires