When does the zeta function take on integer values?

Regarding 3), this "Big Picard" stuff is serious overkill.

Think like an undergraduate real analysis student:

The p-series $\zeta(p)$ converges for real $p > 1$, whereas $\zeta(1)$ = sum of the harmonic series = oo.

An easy argument using (e.g.) the integral test shows that

$$lim_{p \rightarrow \infty} \zeta(p) = 1$$

The function $\zeta(p)$ is continuous in p [the convergence is uniform on right half-planes, hence on compact subsets], so by the intermediate value theorem it takes on every positive integer value $n \ge 2$ at least once -- and, since it is a decreasing function of p, exactly once -- on the real line.

Thus $A_n$ is nonempty for all $n > 1$.

EDIT: Let me show that zeta(s) takes on all real values infinitely many times on the negative real axis.

For this, note that for all $n > 0$,

$$\zeta(-(2n-1)) = - \frac{B_{2n}}{(2n)}$$,

where $B_{2n}$ is the $(2n)$th Bernoulli number. It is known that the $B_{2n}$'s alternate in sign and grow rapidly in absolute value:

$$|B_{2n}| \sim 4 \sqrt{\pi n} \left(\frac{n}{(\pi e)^{2n}}\right)$$

The claim follows from this and the Intermediate Value Theorem.

There are infinitely many roots of $\zeta(s)-a=0$ for every complex number $a$. When $a\ne 0$, these are called "$a$-values" and there is a whole chapter discussing their distribution in Titchmarsh's book on the zeta-function. Selberg also discusses $a$-values in his (now famous) paper "Old and new conjectures and results about a class of Dirichlet series" where he defines the Selberg class.

Here is an overview of some of the important results:

1) There are $\frac{T}{2\pi}\log T + O(T)$ $a$-values of $\zeta(s)$ in the strip $0<\Im s\leq T$.

2) Like the zeros of $\zeta(s)$, Levinson proved that $a$-values cluster near the half-line. That is to say, almost all $a$-values are arbitrarily close to the half-line.

3) Unlike the zeros of $\zeta(s)$, there are provably a lot of $a$-values away from the half-line (though not a positive proportion). Namely, there are $\gg T$ roots of $\zeta(s)=a$ for $a\neq 0$ in any region $A\leq \Re s \leq B$ and $0<\Im s\leq T$ where $A\in (1/2,1)$ and $A$ strictly less than $B$. This is proved in Titchmarsh's book. On the other hand, standard zero-density estimates for the zeta-function tell us that there are $o(T)$ zeros in such a region. Some have suggested that this is evidence for the Riemann Hypothesis.

About (1) I agree with Boris Bukh. There is no conjecture about the location of the a-points of the Riemann zeta function ($\zeta(s) = a$), in the way that we have for the 0-points. And by my next remark, there may well be no reasonable description of the m-points for $m \neq 0$.

About (2) I also agree with Boris Bukh, for a particular reason. There is a universality result for the Riemann zeta function, to the effect that as you move the disk $|s - 3/4| \leq 1/4$ vertically upwards, the Riemann zeta function can be made to approximate arbitrarily well in the sup-norm an arbitrary continuous function on the closed disk that is holomorphic in the open disk. It is just a matter of moving the disk far enough up. Since for an arbitrary holomorphic function there is no connection between the a-points and the b-points for a pair of values $a \neq b$, neither would you expect such a relationship for the Riemann zeta function.

The universality result is due to M. Voronin, see page 308 of the second edition of The Theory of the Riemann Zeta-function by E. C. Titchmarsh. It is crucial to get the second edition, with the end-of-chapter notes by D. R. Heath-Brown. This is the standard reference on the Riemann zeta function, though there is also a very useful book by Aleksandar Ivic.