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][1] 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.

  [1]: https://link.springer.com/book/10.1007%2FBFb0103426