"Has the class already started?"
ています is not just used for making "-ing" forms (start-ing), but also for showing a past transition of state that is still relevant in the present. (The class started earlier, and it is ongoing now.)
かれ·は·けっこんして·います。 He is married. (He became married, and so, is now married.)
かれ·は·はしって·います。 He is running. (This could also mean that "he runs", as a habit.)
I answered the same as the solution, except that I put "already" (もう) in front of started, instead of in front of class. Why would "already" modify class instead of started? And in counting it incorrect, it said the error was at the end (under the す) - although I can see it getting confused.
[Of course, the teaching (web page only) also lists the various forms as including "profibit"!]
「授業は始まっていますか？」"is the class in the state of having started?"
「授業は始まりましたか？」"did the class started?"
I'm pretty sure both can be used, but the latter one can have the nuance that you are not looking for current information, more like "did the class ever started?" or something like that. I think adding もう removes that nuance though. For me, it sounds better to use ている、not sure why though.