“faire” is a bit of a “filler verb” in French. In the second sentence, it clearly means “teach”, or in french: Quel professor donne ce cours de latin?

I’ve never seen it used like in the first sentence, but the meaning is transparent: He’s doing some sort of exercise in Latin. You seem surprised by the structure of that sentence, but it’s exactly the same as the english “Robert is doing maths.”

Faire is a verb that can be used to replace a more specific word

Je fais un gateau –> Je prépare un gateau

Je fais du foot –> je joue au foot

In this case : Robert fait du latin. can be understand like

Robert prend des cours de latin

OU

Robert travaille son latin (Fait des exercices)

We cannot clearly know if he is doing some exercices right now or if he has Latijn lesson.

And for the second part :

Quel professeur fait ce cours de latin?

The verb can be replaced by "Teach"

Quel professeur donne ce cours de latin ?