I have seen $\forall x [P(x)\rightarrow Q(x)]$ written in multiple different forms from different authors. Which of the following are correct if any (as listed below)?
$\forall x [P(x)\rightarrow Q(x)]\equiv P(x)\Rightarrow Q(x)$
$\forall x [P(x)\rightarrow Q(x)]\equiv$ P(x), implies Q(x)
$\forall x [P(x)\rightarrow Q(x)]\equiv P(x) \models Q(x)$
On
As Mauro says: no one! One reason for this that applies to all of them is that $\forall x (P(x) \rightarrow Q(x))$ is a sentence, as it is a formula without any free variables, but in all of the 4 cases you list you are dealing with free variables, i.e. formulas, but not sentences.
Of course, to really see that, you need to know what those different symbols mean, and Mauro explained all that, although one immediate clue should have been that in 1 through 4 there are no $\forall x$ being used.
So, maybe a more interesting question would have been: is the formula $P(x) \rightarrow Q(x)$ equivalent to any of 1 through 4? In which case 3 and 4 are still definitely not the same (see Mauro's answer), while for 1 and 2 they could be the same; by 'implies' we sometimes mean 'logically implies' (which would be like 3), but other times we say 'P implies Q' to mean 'if P then Q', in which case it would be the same as $P \rightarrow Q$. And, as Mauro says, the same is true for $\Rightarrow$.
No one.
$∀x [P(x)→Q(x)]$ reads: "every $P$ is $Q$". See Categorical proposition.
The "syntactic consequence" relation: $P(x)⊢Q(x)$ reads: "from (formula) $P(x)$, (the formula) $Q(x)$ is derivable". See Proof calculus (or Proof system).
The "semantic consequence" relation: $P(x)⊨Q(x)$ reads: "(formula) $P(x)$ logically implies (the formula) $Q(x)$". See Logical Consequence.
$⇒$ is ambiguous; sometimes it is used for (logically) implies (i.e. semantic consequence), sometimes for the connective "if..., then..." (i.e. the conditional: →).