If given a variable $y$ such that $y=f(x)$ can we talk about the value of $y$ for a (separate) number $a$ if we were to substitute the value of $a$ for $x$ in the expression defined as $f(x)$?
Case 1: Would this imply that $y=y_a=f(x)=f(a)$ as $x=a$ for e.g. for every value of $x$ it must be equal to $a$ .
Case 2: Would it be a separate value $y_a=f(a)$ that can differ from $x$ (so it would be 'hypothetical', for example, in theory we would get this value of $y$ if we substituted the value of $a$ for $x$) and could differ from $y$ itself. This will be in fact equal to $y$ and a direct substitution of $a$ for $x$ in $f(x)$ can take place.
Perhaps asking what happens if we take the value of $a$ for $x$ imply that we are substituting $a$ for $x$ and hence receive an expression where $y$ = $f(a)$ for all or some values of $x, a$
Two functions are equal by definition if the have the same domain and codomain and have the same mapping. I suppose what you want to ask is if two functions are the same if whe change the name of a variable in a mapping rule. This would be true. E.g. the functions from $\mathbb R_+\to\mathbb R_+$ defined by the rules $x\to\sqrt{x}$ and $a\to\sqrt{a}$ are equal.
If this is not what you intend, please ask your question in a more intelligible way, so we can understand what you mean.