Well the instructor wasnt wrong when he said you need to look where you want to turn. If you keep your head straight, you wont turn. It helps to complete the twist your body need to make it turn.
1. Turn a transition from heelside to toe side during linking turns i tend to pivot or kick my backfoot out.
2. I tend to just start doing heelside falling leaf on the black runs because i just can't get over how steep it is
My g/f is having the first problem mainly because she doesnt look where her board is going. It makes it hard to complete the turn heelside to toeside. Just try to point with your arm where you wanna go and look there. When it starts to turn (it will for sure) put toes down and there you go!
for problem #2: dont expect to be able to run blacks like it was nothing on your first 5 days of snowboard ever! Start at the bottom line man, green=>blue=>black
The more you'll practice, less will your fear be when you get in terrains that arnt in good shape (fresh groomed)
my 2 cents.