As per the Mahabharat, the chariot given to Arjun by Agni dev was ape bannered i.e. it had a celestial ape on its banner from the very beginning. It was not Hanumanji.
Yes, he should have. In fact, he had earlier in the battle done his duty as charioteer by giving the right advice to Karna to aim his astra lower (the one with Ashwasena). K...