Is the Kleene star of an intersection always equal to the intersection of kleene stars?
Clearly not. 

Let $A=\{a\}$ and $B=\{aa\}$. 

Now, 

$A\cap B =  \emptyset$

so

$(A\cap B)^* = \{\epsilon\}$

but 

$A^*\cap B^*=B^*=\{a^{2i} : i \in \mathbb{N}\}$ 

(all strings consisting of an even number of $a$).