Manipulating Intersection of oracles

Question

Suppose for different classes $A,B,C$ we have that $A\subseteq P^B$ and $A\subseteq P^C$.

1. We have $A\subseteq P^{B}\cap P^{C}$. Does it also mean $A\subseteq P^{B\cap C}$?

2. Supposing $A\subseteq P^{B\cap C}$ also holds does it also give $A\subseteq {B\cap C}$?

3. If $P^B=B$ then does it mean $A\subseteq {B\cap C}$ or $A\subseteq B\cap P^C$?

4. If $B\subseteq P^{B\cap C}$ then does it mean $A\subseteq {B\cap C}$ or $A\subseteq B\cap P^C$?

Do we know anything else about intersection of oracles?