Learn Russian: Они учат этот язык. - They are learning this language.

Normally, yes! Second conjugation verbs usually end in -ят for the "they" form. However, Russian has a spelling rule: after the letter ч (as well as ж, ш, щ, г, к, and х), you can never write я. You must use а instead. That is why it is учат and not учят.

The phrase этот язык is the object of the verb. Why didn't its ending change?

It actually did change into the accusative case, but you cannot see a difference! For masculine nouns that are inanimate (things, not people or animals), the accusative form looks exactly the same as the dictionary (nominative) form. The word язык is masculine and inanimate, so этот язык stays exactly as it is.

I've also seen the verb изучать used for "to study". Could I use it here instead of учат?

Yes, you could say Они изучают этот язык. In Russian, учить and изучать are very similar when talking about learning a language or a subject. Учить is more common in everyday speech, while изучать sounds a bit more formal, like studying something in depth.

Is there a separate word for "are learning" versus "learn"?

No, Russian does not have a separate continuous tense like English does. The single present tense verb учат translates to both "they learn" and "they are learning" depending on the context.