Is $\epsilon$ always contained in $\Sigma^*$?
Normally, $\epsilon$ stands for the empty string: the string with no characters, which would be "" is most programming languages. It's too confusing to have $\epsilon$ be a symbol in the alphabet, so I'm going to rename it and write your alphabet as $\Sigma=\{e,0,1\}$. (The alternative would be to use some other symbol to denote the empty string.)

Now, by definition, $\Sigma^*$ is the set of all finite strings that can be written using the characters of&nbsp;$\Sigma$. This always includes the empty string&nbsp;$\epsilon$ and, as long as $\Sigma\neq\emptyset$, it also contains strings of all finite lengths. So the claim in the question that $\Sigma^*=\{e, 0, 1, 01, 10\}$ is incorrect: $\Sigma^*$ is an infinite set.

The regular expression for $\Sigma^*$ is $(e+0+1)^*$; the automaton consists of a single state, which is accepting and has transitions to itself for each symbol in&nbsp;$\Sigma$.