When are additional Dominion actions chosen?
Simply? Yes.

Imagine you have a little "action bank". At the start of your turn, it contains "1 Action" which you can spend to play an Action card. If you play a terminal card (like Smithy) then you're down to "0 Actions" and you're done. If you play a non-terminal like Market, after it uses up the action in the bank, it gives +1 Action - so it repays itself, and you can play another Action card. If you play a village/splitter that says "+2 Actions", you get to put both those actions in the bank. So you can then use one to play a Smithy, draw three cards, then use the second action to play any Action card in your hand, including one drawn with the Smithy.

"+1 Buy" and "+1 Coin" work in the same way, except that you can't use either of them until your Buy phase (i.e. once you're done playing all your Action cards), with a slight exception with Black Market. "+1/2/3 Cards" is the only one that's different, in that its effect resolves instantly without choice - when you play a card that tells you to draw cards, you draw them immediately.