> Does \\( (X \rightarrow Y) \times B = X \times B \rightarrow Y \times B \\)?

Not in general, no. Not even for any Cartesian Closed Domain.

> Does \\( (X \rightarrow Y)^B = X^B \rightarrow Y^B \\)?

Not in general, no. Not even for any Cartesian Closed Domain.

----------------------

However, in a Cartesian Closed Domain we **do** have:

\\[
(X \to Y)^B \cong X \to (Y^B)
\\]