You are correct, the cable from the brake to the engine is to pull the throttle closed when braking, it's a safety feature built in to some karts; you don't need it but, if it's easy to connect to the new engine, you may as well do so...
AFAIK, those calipers are adjustable but, more than likely, you are again correct, the pads may be work out... As for whee to find replacements, sorry, I can't help there...