Can you apply XOR to the following expression? So I know that 
$$\bar{x}y + x\bar{y} = x \oplus y$$ 
Can this be applied to something like
$$ \bar{a}\bar{b}cd + ab\bar{c}\bar{d} $$
to get 
$$ab \oplus cd$$

(Assuming I'm interpreting your notation correctly: $\overline{x}$ is "not $x$", etc.)
No, because the negation of $ab$ isn't $\overline{a}\overline{b}$, it's $\overline{a}+\overline{b}$. So you could write $$(\overline a+\overline b)cd+ab(\overline c+\overline d)=ab\oplus cd.$$
