"The class has already started."
This is a difficult one.
I think that the word 'has' in the English translation is telling. If you were simply to say 'The class already started' you would be correct.
By 'has started' it is emphasised that it has not only begun, but is currently in session.
じゅぎょうはもうはじまりました = The class already began.
While possibly not grammatically correct, the example sentence would be closest to 'The class is started'.
The "....te imasu" form describes the state of action. In this case, the starting is already underway and that is the way things stand as far as the statement is concerned. The translation could either be "is (start)ing" or "has (start)ed."
English verbs have tense (reference time of action) but Japanese verbs have aspect (reference a state of action as incomplete, complete or static). These categories do not exactly correspond.
"Hajimatte imashita" would mean (more or less literally) "was in a state of starting" or (in more natural English) "had started."
Hope this helps. If not, just ignore it.
Both "hajimari" and "hajimatte" are renyookei forms that link verbs in series. There is no appreciable difference in meaning between them but there tends to be a closer logical relationship between the linked verbs when the "hajimatte" form is used. The relationship can be logically coordinate or subordinate with either form.
Sorry, I didn't realize that DL was also teaching that form of renyoukei.
Both hajimari and hajimatte can also be used as renyoukei, in bridging clauses.
I can't picture a verb directly following hajimari unless it was part of the next (separate) clause.
The Japanese is not derived from English. "Imasu" indicates a current state of affairs. The "-te" form is a conjunctive form (renyookei) that links the action with the stative verb. So, the sense is that "starting (or having started) is the current state of play." Depending on context, this can be translated by the English present perfect or present progressive.