Writing predicates - is the quantifier necessary?

No, we cannot omit the existential quantifier.

How we read the formula:

$x=2m$ ?

As: "every $x$ is even" ? Or as: "some $x$ is even" ? Or as: "every $x$ is the double of every $m$" ? Or as: "some $x$ is the double of every $m$" ?

The "usual" convention is that we can omit the universal quantifier.

If we apply it, we read:

$\exists m \ (x = 2m)$

as: $\forall x \exists m \ (x=2m)$, i.e. as: "every number is even".

---

Regarding the second part of the question, if we write, for $x \ne 1$: $\text{Prime}(x)$ as $\forall a (a|x \to \ldots)$, clearly:

$\text{Comp}(x) \text { iff } \lnot \text {Prime}(x) \text { iff } \lnot \forall a (a|x \to \ldots) \text { iff } \exists x \lnot (a|x \to \ldots)$.

Notice how '$x$ is even' makes explicit reference to $x$, but not to $m$. That should be an indication already that in your formula you want $x$ as a free variable (i.e do not quantify the $x$) but not $m$, so use:

$$\exists m \ x = 2m$$

Also note that depending on how you quantify the $x$, you get a different statement:

$$\forall x \exists m \ x =2m$$

means that all numbers are even, while

$\exists x \exists m \ x=2m$

means that some number is even. (note that in both interpretations there is no longer a reference to $x$, again signaling that the $x$ has been quantified)

Likewise, the meaning of

$$x =2m$$

(which, by itself, reads as '$x$ is twice $m$' ... so now you do have an explicit reference to $m$, reflecting the fact that $m$ is a free variable)

depends on how $m$ is being quantified:

$$\exists m \ x=2m$$

clearly means that $x$ is even, while

$$\forall m \ x =2m$$

clearly means something different!

So yes, you really want to existentially quantify that $m$.

Same story for '$x$ is composite': it makes no reference to any $a$, and so you do need to quantify that $a$ and have only $x$ as a free variable. And also note that if you were to quantify the $a$ existentially, you don't get what you want.

Finally, for '$x$ is composite' it makes sense to just negate whatever you have for '$x$ is a prime' (and hence any quantificational issues will be taken care of automatically) ... if it weren't for the fact that $1$ is neither prime nor composite! So what you really want there is '$x$ is not $1$ and $x$ is not prime'