Knowing that the ball was gold gives you Bayesian knowledge about the boxes behind the door, since the prior probability of the host pulling a gold ball from a 6-gold door is different than from the 3/3 door. So you have to multiply Monty Hall probabilities and Bayesian probabilities together.
That assumes the host pulled a ball at random, of course, and not a deliberately gold ball.
Yes! Cancels out, leaving only a very slight edge on door 2. All that work for only… 2.77% edge over picking at random. What a troll problem, huh?