Is the empty string always in a finite alphabet? Is the empty string always an element of an aribitrary finite alphabet? 
I understand that the empty string is part of the Kleene-Star of any alphabet, but is it intrinsically part of any finite alphabet where I don't explicitly mention it?
For example, if $A=\{a,b\}$, is $\epsilon$, the empty word, in $A$? Or, would it only be in $A$ if it were specified that $A=\{\epsilon,a,b\}$?

The empty string, often called $\varepsilon$ may be created using any alphabet, by creating the string which contain no characters from the alphabet. An alphabet does, by definition, only contain characters which later may be used to create strings. Kleene-star of an alphabet is a language, why it is possible for the empty string to exist in there.
In any alphabet $A$ we have that $\varepsilon\notin A$, this is because the $\varepsilon$ is not a character, but rather a string and specifically the empty string. If you create the language $\{a,b,\varepsilon\}$ then $\varepsilon$ does not represent the empty string but rather is just a character like $a$ and $b$.
Summary: The empty string is by definition not a charcter and thus it is not part of any alphabet.
