A relevant keyword in this context is Sidon set (however Sidon is also used in a different context, so when searching do not be surprised if you stumble also over unrelated things).
A subset $T$ of an abelian group $G$, written additively, in your case $\mathbb{Z}/m \mathbb{Z}$ with addition, is called a Sidon set if for $a_1,a_2,a_3,a_4 \in G$ one has
$a_1+a_2 = a_3+a_4$ if and only if $\{a_1,a_2\} = \{a_3,a_4\}$ (the sets), in other words
$(a_1,a_2)$ or $(a_2,a_1)$ is equal to $(a_3,a_4)$ (the pairs). Or, verbally, their is no non-trivial way that two sums can be equal (the trivial one being of course that the summands are just flipped).
In yet another way, a simple upper bound for $|T+T|$ is, for $|T|=k$, the expression $k(k+1)/2$, arising as ($k$ choose $2$) plus $k$ [the sum of the elements of two-element subsets plus the 'symmetric sums'] or equivalently $(k^2 - k)/2 + k$ [half the number of pairs with distinct entries plus those pairs with twice the same entry].
Then, a Sidon set is a set that attains this upper bound. For a Sidon set $|T+T|$ is maximal among all elements of this cardinality, and one knows the cardinality precisely.
This upper bound also answers one of your questions. For $k \le \sqrt{M}$, one has the upper bound for $|T+ T|$ of the form $(M + \sqrt{M})/2$ which is less than $M$ (except for $1$).
So for $k \le \sqrt{M}$ you can never get all elements (except for $M=1$).
Thus to some extent your question is really about Sidon sets. However, if $k$ is too large relative to $M$ the problem arises that no Sidon sets of that cardinality exists anymore.
For very large $k$, the problem becomes 'cheap'. For example if $k > |G|/2$, then you have that $T\cap (g - T)$ is nonempty for each element of $g$ of $G$ and, thus, since $a_1=g-a_2$ with $a_i \in T$ one has in fact $g \in T+T$ and $T+T$ is the full group, and of course cannot be larger.
Since this argument works for each set this can of course be improved (considerably).
A key-word here would be 'additive basis of order two'. I do not continue on this as it seems to me, you care more about small $k$ (in a relative sense).
Regarding your second question, on construction, this is an open problem (towards which however various results are known). The problem is that the precise maximal cardinality of a Sidon set is not known. So you do not know up to which exact value the bound $(k+1)k/2$ will actually be the right value.
However, there are good constructions for Sidon sets known, and perhaps this is sufficient for you. For a detailed overview of results on Sidon-related resuls see Kevin O'Bryant's annotated bibliography (many mant references). Note however that it is mainly on results in the integers (Sidon sets in the first $n$ integers, or infinite ones, then often called Sidon sequences). Yet, it also contains a section on cyclic groups and in part the results on integers are relevant too.
