There is no complete classification, but some structural results are known. To give you something to search for: such groups are called $\mathbb{Q}$-groups. There is a whole book devoted to their structure: Structure and Representations of $\mathbb{Q}$-Groups by D. Kletzing. You will find there answers to many, if not all, of your questions.
In particular, the converse of the lemma is indeed true. It follows from two facts: firstly, characters "separate" conjugacy classes, i.e. if two elements are not conjugate, then there exists an irreducible character that takes different values on them; and secondly, if $m$ is coprime to the order of $g$, then for every irreducible character $\chi$ the values $\chi(g)$ and $\chi(g^m)$ are Galois conjugates.