How can a proper subset be a subset of itself?

Because every set is a subset of itself. A proper subset cannot, however, be a proper subset of itself.

yes

If set A is a subset of set B, that means that all elements in set A are also in set B. In the case of a proper subset, there is the additional specification that the two sets are not equal, i.e.,...

A subset of another set is that set which contains all the elements of the other set. The 2 sets may or may not be equal. A proper subset of another set is that set which contains all the elements of...

Assume that set A is a subset of set B. If sets A and B are equal (they contain the same elements), then A is NOT a proper subset of B, otherwise, it is.