Showing the following sequents are not valid
First, $\phi \models \psi $ means that $\psi$ is a consequence of $\phi$. In other words, if $M$ is a structure and if $M \models \phi$ ($M$ models $\phi$) then $M \models \psi$ ($M$ also models $\psi$).

Now, regarding your statement $$\lnot \forall x P(x) \models \forall x \lnot P(x)$$
clearly $\forall x \lnot P(x)$ is not a consequence of $\lnot \forall x P(x)$. To show it consider the set of natural numbers as a structure $M$ and $P(x)$ as the predicate "$x$ is divisible by $2$". Then 
$$\lnot \forall x P(x)$$
means "Not for all $x$, $x$ is divisible by $2$ which is a true sentence (since in fact odd numbers are not divisible by 2), while 
$$ \forall x \lnot P(x)$$ 
means "For all $x$, $x$ is not divisible by $2$" which is not a true sentence (since in fact even numbers are divisible by $2$). 

So, the structure $M$ (the set of natural numbers) is an example showing that $M \models \lnot \forall x P(x)$, but $M \nvDash  \forall x \lnot P(x)$ and your statement 
$$\lnot \forall x P(x) \models \forall x \lnot P(x)$$
is not true.