Q: Can a player take income for his districts in two parts, before and after building?
A: No. You can take these revenues when you want, but all at once. So, a player can either take the incomes of his districts before constructing new districts (if he has need of money in order to construct), or after having constructed (if the new district give him income).
On the other hand, district income is completely distinct from the two golds you can get at the beginning of your turn, instead of drawing a card.The merchant extra gold piece is not a district income and is also taken at the beginning of the turn So you can take two golds, build, and then get income from your districts, but you can't get income from some districts, build, and then get income from the district you've just built. .