Question Regarding the Replacement Schema For each formula $\phi(x,y,p)$, we have the following axiom:
$\forall x \forall y \forall z (\phi(x,y,p) \wedge \phi(x,z,p) \rightarrow y = z) \rightarrow \forall X \exists Y \forall y (y \in Y \leftrightarrow (\exists x \in X) \phi(x,y,p))$
This set of axioms has been characterized informally in English as follows: 
"If a class $F$ is a function, then for every set $X$, $F(X)$ is a set."
But isn't the English statement saying much less than what the actual schema is saying?  That is, it seems that the schema can be interpreted as saying what the English statement is saying only if we interpret $\phi(x,y,p)$ as being the property of "$y$ is the image of $x$ under $p$".  But then aren't there many more interpretations of $\phi(x,y,p)$ that might permit this axiom schema to be saying some radically different things than this?  In any event, how should one interpret formulae of the sort $\phi(x,y,p)$ as above?

I don't know why you think it only makes sense with "$y$ is the image of $x$ under $p$".
The first part $\forall x \forall y \forall z (\phi(x,y,p) \wedge \phi(x,z,p) \rightarrow y = z)$ defines a class function $F_p$ by saying $F_p(x)=y$ whenever $\phi(x,y,p)$ holds.
The second part
$\forall X \exists Y \forall y (y \in Y \leftrightarrow (\exists x \in X) \phi(x,y,p))$ then says there is a set $Y$ such that $Y=\{y : \exists x,\phi(x,y,p)\} = F_p(X)$.
