I would guess 始めたということです is supposed to say '' It means they started talking on the phone by saying Oi Oi '' whereas the sentence with 始めました means '' They started talking on the phone by saying Oi Oi ''.
And for the 2nd example, I'd personally use them both equally, but I think the 2nd one sounds more natural.
For the last (second) question, the subject is '' they / people in general ''
So the sentence means '' They used to say moushimasu moushimasu ''
Sorry if this answer is not organized well. Hope this helps.
ganbatte kudasai (^^)/