[ste_lazzaro]

Hi,
Yes, A B C on the ESC shall be connected to A B C on the motor.
Most people solder both ends, but if you prefer you can put a connector (tube-banana) somewhere in between, on each of the leads.

The "black bit of plastic" is a condenser. The ESC needs it so leave it as it is.
You will have to solder the battery + and - to the same posts.

The small connector on the battery is for the balance charging.
You should have purchased a charger for your Lipos, featuring balance charge mode.
It should have a similar connector (female), and hopefully some instruction on how to balance charge your battery.

The leads coming with the battery are supposed to be used between battery and charger. You don't have to solder it to the ESC as you would end up having the balancing leads on the ESC (useless).

Hope this helps!