Can I have a "dependent coproduct type"?
The dependent sum is a common generalization of both the cartesian product $A \times B$ and the coproduct $A + B$. It just so happens that the HoTT book introduces dependent sum by generalizing $A \times B$, because that does not require the boolean type to be defined first.

The coproduct is a special case of dependent sum. Given types $A$ and $B$, consider the type family $P : \mathtt{bool} \to \mathcal{U}$ defined by $P(\mathtt{false}) = A$ and $P(\mathtt{true}) = B$. The dependent sum $\sum_{b : \mathtt{bool}} P(b)$ is equivalent to $A + B$. By the way, you can also get $A \times B$ as a dependent product $\prod_{b : \mathtt{bool}} P(b)$.

You ask what is special about products and function types. There are many, many reasons why $\sum$ and $\prod$ are "necessary". In terms of logic, they are necessary because they correspond to $\exists$ and $\forall$ by the propositions-as-types correspondence (but that only shifts the question to "why are $\exists$ and $\forall$ necessary?"). In terms of category theory, $\sum$ and $\prod$ are necessary because they are the left and right adjoint to substitution. Ask a more secific question if you would like to know more.