Short answer: Such a set $S$ is known as a **restricted additive basis**.

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As observed in comments, the general term to search for is *[finite]
additive basis*: a finite set $S \subset \mathbb{N}$ such that $S+S
\supseteq [2n]$.  Another term is *[postage stamp problem][1]* (not to
be confused with the Frobenius postage stamp problem which is
different).

The extra condition that $S \subseteq [n]$ makes this the
*restricted postage stamp problem* and the solutions are *restricted
additive bases*. ("Restricted" means that the elements are restricted
to be in $[n]$.)

It is then (almost) equivalent to ask
 1. given $k$, what is the largest $n$ ...
 2. given $n$, what is the smallest $k$ ...

such that there exists a restricted additive basis of $k$ elements for
the interval $[2n]$.  More about that "almost" at the end of this
answer.  Let us write $n(k)$ for the largest $n$ when $k$ is given,
and $k(n)$ for the smallest $k$ when $n$ is given.

Here the question asks for $k(n)$.  For $n(k)$ this is OEIS
[A006638][2], where currently the largest entry is $a(47)=734$ by
your truly.  Some translation needed: That "47" counts only the
nonzero elements, so in fact there are 48 elements; that "734"
means the target interval is $[2n]=[734]$, so in our current notation
it becomes $n(48)=367$.

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To the specific questions:

**Q1.** The OEIS entry gives some pointers to literature.
The oldest reference that I know is H. Rohrbach, "Ein Beitrag zur additiven
Zahlentheorie", *Math. Z.*, 42 (1937), 1-30, which studies both the
unrestricted and the restricted versions.
Symmetric bases have also been studied, see for example S. Mossige,
"Algorithms for computing the h-range of the postage stamp problem",
*Math. Comp.* 36 (1981), 575–582.

**Q2.** About asymptotics: No, one cannot achieve $k(n) \sim
2\sqrt{n}$ asymptotically.  Gang Yu, "Upper bounds for finite additive
2-bases", *Proc. AMS* 137 (2009), 11-18, shows that

$$\limsup_{n \to \infty} \frac{n}{k_r(n)^2} \le 0.419822,$$

where $k_r(n)$ is the smallest size of a restricted basis for $[n]$.
I believe this is currently the best lower bound for $k_r$.
Translated to our notation (with target interval $[2n]$)
this means the factor in front of $\sqrt n$ is asymptotically at least $2.18264$.

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And about the "almost": Generally $k(n)$ increases as $n$
increases, but there are some known cases where $k(n)$ suddenly steps
down. One of them was also noticed in the comments above: "What the
heck is going on at n=31?".  Indeed we have $k(31)=14$ but $k(32)=13$;
$k(51)=18$ but $k(52)=17$; and $k(57)=19$ but $k(58)=18$.
See Table 5 in [this paper][3] by me and coauthors, where the
corresponding problem is studied in two dimensions.


  [1]: https://mathworld.wolfram.com/PostageStampProblem.html
  [2]: https://oeis.org/A006638
  [3]: https://cs.uwaterloo.ca/journals/JIS/VOL21/Rajamaki/raj.html