Baker Gill Solovay $P^B \ne NP^B$ relativization, what class is $B$ in?
For question 1, the BGS construction can be performed in exponential time, so you can construct such $B \in \mathsf{EXP}$.

(For question 2: Sasho Nikolov's answer was originally only for $\mathsf{\Sigma_k P}$-complete languages, and I pointed out that one can also take any $B' \in \mathsf{NP} \cap \mathsf{coNP}$, since $\mathsf{NP}^{\mathsf{NP} \cap \mathsf{coNP}} = \mathsf{NP}$. But Sasho's updated answer subsumes this case.)