The following question from MathOverflow should answer your questions and more: Smallest permutation representation of a finite group?
The answer to 3) is yes, and more generally if $G$ is a finite abelian group such that
$$G \cong \mathbb{Z}_{p_1^{a_1}} \times \cdots \times \mathbb{Z}_{p_t^{a_t}}$$
where $p_i$ are prime and $a_i \geq 1$, then the minimal $n$ is $p_1^{a_1} + \cdots + p_t^{a_t}$. A proof can be found in the following paper that Jack Schmidt mentions in his MO answer.
Johnson, D. L. "Minimal permutation representations of finite groups." Amer. J. Math. 93 (1971), 857-866. MR 316540 DOI: 10.2307/2373739.
Finding subgroups of $S_n$ isomorphic to $G$ is the same thing as finding injective homomorphisms from $G$ to $S_n$.
Finding homomorphisms from $G$ to $S_n$ is the same as choosing conjugacy classes of subgroups $H_i$ of $G$ so that $n = \sum [G:H_i]$. The kernel of the homomorphisms is the intersection of the conjugates of all of the $H_i$.
If $G=H\times K$, then take $H_1 = H \times 1$ and $H_2 = 1 \times K$, and we get $H_1 \cap H_2 = 1 \times 1$, so the homomorphism defined by $\{H_1, H_2\}$ is injective, and $G$ is a subgroup of $S_n$ for $n = [G:H_1] + [G:H_2] = |K| + |H|$.
Repeating this with 3 groups shows that $C_2 \times C_2 \times C_2$ is isomorphic to a subgroup $S_6$ since $6=2+2+2$. By considering a Sylow 2-subgroup of $S_5$, one can see that $6$ is in fact the smallest possible $n$. An explicit subgroup is $\langle (1,2) \rangle \times \langle (3,4) \rangle \times \langle (5,6) \rangle$.
Similarly $S_3 \times S_3$ is isomorphic to a subgroup of $S_6$ since $6=3+3$. It is not isomorphic to a subgroup of any smaller $S_n$ since $S_5$'s order is not divisible by 36. An explicit subgroup is $\langle (1,2), (1,2,3) \rangle \times \langle (4,5),(4,5,6) \rangle$.
$Z_9$ is isomorphic to a subgroup of $S_9$, and by checking a Sylow 3-subgroup of $S_8$, one can see no smaller $n$ will work. An explicit subgroup is $\langle (1,2,3,4,5,6,7,8,9) \rangle$.
See https://mathoverflow.net/questions/48928/smallest-n-for-which-g-embeds-in-s-n for some higher level ideas.
Best Answer
It is probably worth noting that embeddings in symmetric groups of minimal degree have been studied by multiple authors, amongst them, D. L. Johnson, Minimal permutation representations of finite groups, Amer. J. Math. 93 (1971), 857-866, D. Wright, Degrees of minimal embeddings for some direct products, Amer. J. Math. 97 (1975), 897–903. See also N. Saunders, Minimal Faithful Permutation Degrees of Finite Groups, Aust. Math. Soc. Gazette 35, no.2 (2008), 332-338, and Strict inequalities for minimal degrees of direct products, Bull. Aust. Math. Soc. 79, no.1 (2009), 23–30 by the same author.
Further, and to be complete, I should mention the paper by David Easdown and Cheryl Praeger, On minimal faithful permutation representations of finite groups, Bull. Austral. Math. Soc. 38 (1988), 207-220, and a more "recent" paper that bears the same title, but was authored by L.G. Kovács and Cheryl Praeger, Bull. Austral. Math. Soc. 62 (2000), 311-317.
Finally, in Minimal embeddings of small finite groups (see https://arxiv.org/abs/1706.09286) Robert Heffernan, Des MacHale and Brendan McCann determine the groups of minimal order in which all groups of order $n$ can embedded for $1 \leq n \leq 15$. They further determine the order of a minimal group in which all groups or order $n$ or less can be embedded, also for $1 \leq n \leq 15$.
One nice result is the following: Let $G$ be a group of minimal order in which all groups of order $12$ can be embedded. Then $G \cong S_3 \times S_4$.