Decidability of prefix language
Note that using an existential quantifier in front of a decidable language we can obtain any r.e. language, i.e. every r.e. language is expressible as 

$$ \{ x\in \Sigma^* \mid \exists y \in \Sigma^* \ \langle x,y \rangle \in V \} $$

where $V$ is a decidable language. These include undecidable r.e. languages like 
$$A_{TM} = \{ \langle e, x\rangle\ \mid \text{$e$ encodes a Turing machine which accepts $x$} \}$$. 

The only difference here is that here we have to separate $x$ and $y$ ourselves. The standard trick is to use a new symbol to separate the two parts (assume that the separator belongs to $y$). Therefore any r.e. language including undecidable ones can be expressed in this format.

For the second question, there is no general algorithmic way to check if the prefixes of a  given decidable language is undecidable. This follows from Rice's theorem.