Clearly, there are examples for the second question. Each Hausdorff abelian paratopological group (that is, a group endowed with a topology making the multiplication continuous) which is not a topological group (for instance, the Sorgenfrey arrow, that is the real line endowed with the Sorgenfrey topology generated by the base consisting of half-intervals $[a,b)$, $a<b$) is a counterexample. Less trivial and locally compact counterexample (which necessarily is not a group, because each locally compact paratopological group is a topological group) should be the additive semigroup of non-negative real numbers, endowed with the standard topology and then with isolated zero. (Maybe I even wrote a proof somewhere).
I don’t know (yet) an example for the first question. I looked through the following papers, but without any suggestions:
Francis T. Christoph, Jr. Embedding topological semigroups in topological groups, Semigroup Forum, 1 (1970), 224-231.
A. Mukherjea, N.A. Tserpes. A note on the embedding of topological semigroups, Semigroup Forum, 2 (1971), 71-75.
Sheila A. McKilligan. Embedding topological semigroups in topological groups, ?, 127-138
So I’ll going to think about this question and I also have suggested to think our specialists in topological algebra.