Nope, nowhere near as deep. Most of the math you do in the other subjects are pretty specific and you can revise pretty quickly (like graphics is only about matrix algebra, and pretty straightforward ones at that)
Do you want a life?
I did higher comp and higher phys when I started. Didn't do higher math - I don't have a thing for maths. I think it was alright. Phys can be a bit tough at times, and higher comp is very enjoyable.
Swap the one of the two clashing classes for itself, then you'll be able to pick different streams of lectures/tutes/labs to find a timetable that hopefully doesn't clash.
http://community.boredofstudies.org/589/general-university-new-south-wales/235424/unsw-id.html
You will memorise your student number, or you will be shot.
My blanket advice for normal vs higher:
If you're the academically capable type, you can handle your work and get off your ass, then go higher.
Normal/higher courses have leeway (sometimes even past census) for students to go between the classes if they find they're not suitable.