Is it true that when a magnet attracts something, that other thing does not have to be a magnet?

Correct. The only thing that matters for the non-magnet object is that the electrons must be able to travel through it with relative ease (which is why you won't see a magnet sticking to a tree, for example).

a magnet can potentially attract a non magnet, but cannot repel it?

Also correct, for generally the same reason as the first question but with more detail. When you're placing a magnet on your fridge, it is either attracting local electrons in order to stick, or repelling local electrons leaving a local positive charge, which will still lead to the magnet sticking.

does it matter if you use the + or - charged end of the magnet to attract it?

Nope, either one should work.