School rank? Teacher quality? Previous cohorts performance in HSC externals (Check out Topscores?) Rankings?
I usually recommend students stick with 12 units, at least for the first term.
I've been to new residence and would recommend it if you are wanting to cook/live fairly independently. I thought the rooms, plus kitchen + bathrooms were nice and I'd be happy to live there.
I think your teacher advised you guys poorly unfortunately although it does sound like something a lot of teachers wouldn't have taught their students.
Three unit bridging courses are available or once you've finished HSC you can start learning the level of mathematics required (I also recommend you do the bridging course).
If we are talking about question 30C) (The one with the school playground) I agree that we definitely can't use Pythagoras and should use a different formula.