Families of sets indexed by another set
The notation $\{ x_1,x_2,x_3,\ldots,x_n \}$ means &quot;the set of elements $x_1,x_2,x_3,\ldots,x_n$&quot;. In particular, all elements in the set are listed. If an element isn't listed, it doesn't belong to the set. For example, $\epsilon$ is not an element of $\{ab,b\}$. There is absolutely nothing special about $\epsilon$. It is a bona fide word, and undergoes no special treatment. It is not an element of $\{ab,b\}$ just as $abba$ is not an element of $\{ab,b\}$.
As for your second question, here is a family of sets indexed by $I = \{a,b,c\}$:
$$ B_a = \{a,ba\}, B_b = \{b,ba\}, B_c = \{\epsilon,aa,bbb\}. $$
The union of this family is
$$ \bigcup_{i \in I} B_i = B_a \cup B_b \cup B_c = \{\epsilon, a, b, aa, ba, bbb \}. $$
The set $I$ is completely arbitrary. It could be the empty set, it could be a singleton, it could be a larger finite set, it could be an infinite set. For example, we can define a family of sets indexed by the natural numbers (including zero): for every $n \in \mathbb{N}$, $S_n = \{n,-n\}$. The union of this family consists of all integers: $\bigcup_{n \in S_n} S_n = \mathbb{Z}$,