> If every element of A is present in Set B, then A is a subset of B

Yes, this is exactly the definition of a subset.

> Does this hold true only when the cardinalities of A and B are the same?

No.

> If A = {1,2,3} and B = {2,1,3}, can I say that A and B are equal sets?

Yes.

> If so, does it mean that the order of the element in the set does not matter and only the existence of that particular element does?

I would phrase this even more strongly: there is no ordering of the elements in a set at all. Of course when we write the set down we have to choose some ordering, but that ordering isn't inherent to the actual set in any way. Really {1,2,3} and {2,1,3} are just different names for the same mathematical object, just as 1/2 and 2/4 are different names for the same object.

> if A is a proper subset of B, can we say (without looking at the elements of the sets themselves) that n(A) < n(B)?

As long as A and B are finite, yes.

> "Let A be a set then the null set is a subset of A." I can't seem to wrap my head around this; Google says it's an axiom and axioms can't be proven.

It is not an axiom, but rather follows from the definition of a subset. The empty set has no elements, and so trivially all of those elements are also elements of any other set A that we are comparing it to. For intuition, it may help to think of subsets in the alternate but logically equivalent sense: a set A is *not* a subset of the set B if there is an element of A which is not in B.