Is every Closed set a Perfect set?

No finite set is perfect but every finite set is closed; a finite set has no limit points and thus all of its limit points (all zero of them) belong to it, so it's closed.

Closed means all limit points are in E. But that doesn't mean all points in E are limit points. Any closed set with a point that is not a limit point will not be perfect.

The easiest counter example is a set with a single point. That set is closed but its one point isn't a limit point.

Less trivial and less contrived is $D = \{a + 1/n| a \in \mathbb Z,n \in \mathbb N\} $. Every integer is a limit point. No other point is a limit point. All integers are in D (because $a + 1/1$ is an integer) so D is closed. But for all $n > 1$ then $a + 1/n $ is in D but is not a limit point. So D is not perfect.

No. Consider the set $[0,1] \cup \{2\} \subseteq \mathbb{R}$ with the usual topology. By the definition of limit point, if $x_0$ is a limit point of the set $S \subseteq \mathbb{R}$, $$ \forall r > 0, \exists x \in \mathbb{R}\mid x \neq x_0, |x-x_0| < r $$ Clearly the point $2$ does not satisfy this condition(taking $r = 1/2$), and thus is not a limit point.