Question: Can epsilon be in the input alphabet of an FST ?

Answer: 
The alphabet $\Sigma$ of an automaton can be *any* nonempty set of finite symbols. Surely Greek-speakers would be upset if we forbade $\{\alpha, \beta, \dots, \epsilon, \dots, \omega\}$ as a valid alphabet! If you want, you can use $\{\clubsuit,\diamondsuit,\heartsuit,\spadesuit\}$ as your alphabet. Or $\{A, \alpha, \clubsuit,8\}$. Or any other finite, nonempty set.

However, certain symbols have special meanings and we tend not to use them in automaton alphabets. For example, if you want your alphabet to consist of the symbols $\}$ and $\{$, things get a bit awkward: you start writing $\Sigma = \{\},\{\}$ and, er, yeah, that doesn't quite work. Likewise, it's confusing if $\Sigma$ contains symbols such as ${}^\*$, $($, $)$, $+$ and so on: you can easily write $\Sigma = \{{}^\*, (, ), +\}$ but, now, when you try to write regular expressions over that alphabet, it's impossible to tell which characters in the regular expression are symbols from $\Sigma$ and which are operators in the regular expression.

$\epsilon$ fits into this second class. When $\epsilon$ isn't in the alphabet, we use that symbol to denote the empty string. As such, if you include $\epsilon\in\Sigma$, it becomes unclear whether writing "$\epsilon$" means "the empty string" or "the string containing one symbol, which is the Greek equivalent of 'e'". So, as a practical matter, we usually prefer not to use $\epsilon$ as a symbol in the alphabet. If you do need to include $\epsilon$ in your alphabet, you should use some other symbol to denote the empty string and you should say so. Similar reasoning applies to all other symbols that could be confused in this and other ways.
Basically, the epsilon (denoted by $\epsilon$) is a string, namely *[empty string](https://en.wikipedia.org/wiki/Empty_string)* (altrenatively denoted by $\lambda$) cannot be a part of an [alphabet](https://en.wikipedia.org/wiki/Alphabet_(formal_languages)) since a string itself consists of finite sequence of elements of the alphabet, i.e., *symbols*. Empty string is a string of length zero and so it is not a symbol by definition. 

Furthermore, by the formal [definition](https://en.wikipedia.org/wiki/Nondeterministic_finite_automaton#NFA_with_.CE.B5-moves) of finite automata, the input alphabet is defined as a finite set of symbols, i.e., not strings and hence the empty string cannot be an input symbol. 

However, epsilon is allowed as a part of the input in [generalization of NFA](https://en.wikipedia.org/wiki/Nondeterministic_finite_automaton#NFA_with_.CE.B5-moves), called *NFA with $\epsilon$-moves*. In this case the transition function is defined as a function from $\Sigma \cup \{\epsilon\}$ into $2^Q$. But epsilon is still not a member of alphabet it is just a part of definition of the transition function.