Is every element on a set also a set?

In the context of axiomatic set theory, we (usually) only allow sets whose elements are sets. So in that context, there is no such thing as "the set of all the letters of the alphabet", or your other examples. However, that doesn't mean you can't talk about these concepts in axiomatic set theory--you just have to encode them in sets. For instance, instead of talking about the letters of the alphabet, you could talk about the set of natural numbers from $1$ to $26$, with the understanding that $1$ secretly stands for A and $2$ secretly stands for B and so on. This isn't really any different from the idea of "constructing" numbers as sets which you said you were familiar with: we have an intuitive idea of numbers, and we find a way to talk about them using only sets whose elements are all sets. Similarly, we can find such a way to talk about letters, or classmates, or whatever else you want to talk about that is not intuitively a set.

That depends on which axioms or system you're using. You could of course make up a system which allows (distinct) atomic objects (they're called ur-elements) that can be elements of sets, this is how ZF set theory originally started.

However in (modern) ZF set theory the axiom of extensionality basically prohibits anything that's not a set. Things are considered equal if they have the same elements and since anything not being a set does not have any elements they would be considered equal to the empty set.

As you pointed out one constructs the natural numbers and so on by constructing set. However one normally does not use those properties of them (being sets), you almost never see things like $1 \cup 2$ or $0 \in 1$. One should note that the way natural numbers are constructed is not standardized, that is there is different ways to achieve the same (standardized) properties of the numbers (which makes using numbers as they are sets being non-standard and have no universally accepted meaning).