When does BPP with a biased coin equal standard BPP?
1) Yes, but only because of your definition. Take a unary language $L\in EXP\setminus BPP$ (yeah, I know this might be empty, in that case just take something even bigger than $EXP$), that is very sparse in the sense that $n\notin L$ if $n$ is not a tower of $2's$, i.e., of the form $2^{2^{2^\ldots}}$. Define $p=\sum_{n\in L} 1/n$. This $p$ is not $BPP$-computable, but $p$ can be approximated in $P$ up to a small enough additive error that allows the simulation of a $BPP_p$ machine.

Had you defined $BPP$-computable such that you want to approximate $p$ up to an additive error of $1/n$ (instead of $1/2^n$) in polynomial time, things would be different.

Update. The below answer is for the case when the additive error we allow is $2^{-n}$ instead of $2^{-n-1}$.

2) Yes, because here you can forget about the polynomial restriction on the classes and by sampling $2^n$ times you can get the $n$-th bit of $p$ in $BPP_p$.