Exercises/is exercising would be заниматься (you can add спортом or физкультурой).
Yes, the English doesn't sound right. "Trains" is usually used as a transitive verb, therefore you could say "my sister is training the swim team." We have an expression, "my sister is in training" if she is preparing herself or being prepared to compete in a sport.
I looked it up and it is possible to use it as an intransitive verb as they do in the "solution," but it is not used often and sounds wrong to my ears. It really depends on the context, and here we have none.
I think, "моя́ сестра́ практику́ется" is a beter translation for "my sister is practicing", while "моя́ сестра́ трениру́ется" should be translated as "my sister is training".
The infinitive is: Тренироваться. Conjugation charts for this and all verbs are on Wiktionary/Vikislovar' - https://en.wiktionary.org/wiki/%D1%82%D1%80%D0%B5%D0%BD%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D1%82%D1%8C%D1%81%D1%8F (sorry for the unwieldy link).
This conjugation is typical for verbs that have the -ова- / -ева- "suffix" in them - the -ова- / -ева- morphs into -у- (not 100% for verbs spelled that way but it is common).
"just" is not part of the Russian sentence. You've added a word that isn't there.
If the ся in тренируется means "themselves", isn't "My sister is training herself" still correct?
Technically yes, though in English you don't need the reflexive pronoun because "train" by itself is transitive or intransitive. I wouldn't say it's wrong just not necessary, whereas in the Russian you do need the reflexive suffix because otherwise the verb is exclusively transitive, if that makes sense.
моя сестра тренируется в скрипке. хорошо? или она практикует скрипку? она тренируется в теннис?
What activities does "practice" cover?
In English, for instance, one "practices the piano", "practices medicine' (does the work of a trained physician), lawyers "practice" law (work as lawyers), but "train" is not used for any of these phrases.
"training" to be a doctor means that you're not yet a doctor, while "practicing medicine" means you've received all the training you need and have been certified by Medical authorities to tend to patients.
"Тренируется" most often than not means she's doing physical exercises. In general it's more about trying to become better in some physical activity. It can't possibly mean "practicing medicine", but neither it's "training to be a doctor". It can be "training performing a surgery on a dummy" or something like that. But that requires context, obviously.