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](http://www.franko.lviv.ua/faculty/mechmat/Departments/Topology/).