Overriding the prefix/postfix operators

[D_Drmmr]

Okay...to be honest, I'm not really sure what most of that means.

Well that's what learning is all about. However, a bit of googling will often get you a long way. E.g. C++ explicit constructor

Some of the other problems you listed are, I think, required by my instructor even if they are bad design. Maybe I'm wrong though.

Well, even if your instructor requires you to do things that are considered bad programming, you can still learn them the right way.
Just think about how you would use the class that you've implemented in a program.
However, your teacher did implicitly refer to one of the points I made.

For full credit use const wherever appropriate"

That is called "const correctness".