What's the reason for wiring a relay this way?
There is absolutely nothing strange there - it's just how they've drawn it that's strange.  Let me re-draw it as a proper schematic for you:







simulate this circuit &ndash; Schematic created using CircuitLab

When the switch turns on the relay activates and power flows to the motor (depicted as a lamp here - circuitlab doesn't have a motor).

Why did they wire the battery to pin 87 and not pin 30? Simply because then the motor would be on all the time and the relay would do nothing.

Why did they use the same gauge wire?  Simple economy of scale.  They're already using that gauge elsewhere, so why fork out on a different, smaller, gauge as well when they don't need to?