I don't think such a term is used in set theory. A proper subset, on the other hand, is a subset of the set, that is not equal to the set itself. The difference is comparable to the difference between "greater than" and "greater-or-equal", for real numbers.
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No. The null set cannot have a proper subset. For any other set, the null set will be a proper subset. There will also be other proper subsets.
A set with only one element in it. The only proper subset of such a set is the null set.
The empty set.
NO
It is a set that is well defined.