Why not "most kezd a nyár"?
Kezd is frequently a transitive verb, and has to have some consequent phrase: someone (the subject) begins something, or begins to do something. Kezdődik is always intransitive and means the subject itself is in the state of beginning.
Is "the summer starts now" also acceptable?
Yes, it should be accepted.
Now, the summer is starting was marked wrong. ???
"The summer is beginning now" is wrong? Could someone explain, please?