Relay activated with different power source

If you really using the relay you linked to: that is a bistable relay, which means that current through the coil in one direction switches it to one position, and you need current through the coil in the other direction to get it in the other position.

The behavior that you seem to expect is that of a normal, monostable relay.

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 – 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?