This is not for homework, and I am not totally convinced I understand the question entirely. Also, I'm not allowed to use the fact that $\mathbb{Z}[x]$ is a UFD. The question asks
Show that $f(x) \in \mathbb{Z}[x]$ is irreducible if and only if $f(x)$ is either a prime integer or an irreducible polynomial in $\mathbb{Q}[x]$ such that the $\gcd$ in $\mathbb{Z}$ of the coefficients of $f(x)$ is $1$.
The part of the question I am finding confusing is the phrasing "...or an irreducible polynomial in $\mathbb{Q}[x]$ such that the $\gcd$ in $\mathbb{Z}$ of the coefficients...". What is the $\gcd$ in $\mathbb{Z}$ of the coefficients of a polynomial in $\mathbb{Q}[x]$? Does the author mean to multiply $f(x)$ by an integer $c$ such that the coefficients of $cf(x)$ are integers, and then look at the $\gcd$ of those coefficients? Under this tentative assumption of what the question is asking, here is what I have so far.
If $f(x)$ is a prime integer or an irreducible polynomial in $\mathbb{Q}[x]$ such that the $\gcd$ in $\mathbb{Z}$ of the coefficients of $f(x)$ is $1$, then $f(x)$ is irreducible in $\mathbb{Z}[x]$ (this follows from a theorem that states: "a polynomial $f(x)$ factors as a product of polynomials of degree $m$ and $n$ in $\mathbb{Q}[x]$ if and only if $f(x)$ factors as a product of polynomials of degree $m$ and $n$ in $\mathbb{Z}[x]$").
Conversely, suppose $f(x)$ is irreducible in $\mathbb{Z}[x]$. If $\deg (f(x)) = 0$, then it is not hard to see that $f(x)$ must be a prime integer. If $\deg (f(x)) > 0$, then I can again use the result quoted in the last paragraph to say that $f(x)$ cannot be written as the product of polynomials of lower degree in $\mathbb{Q}[x]$ (for if it could, then $f(x)$ would be reducible in $\mathbb{Z}[x]$). At this point, I'm not sure how to show that the $\gcd$ in $\mathbb{Z}$ of the coefficients of $f(x)$ is $1$. I'm also a bit uneasy about the approach in general. Critiques and hints are very welcome!
The question asks about irreducibility in $\mathbb{Q}[x]$, but the polynomial itself is in $\mathbb{Z}[x]$. So there is always a g.c.d., without anything special happening.
As for your last question: how about a proof by contradiction? What would happen if the g.c.d. of the coefficients were not $1$?