>>>Does anyone know what the MO Mesorah is? Or how old it is? Who started it?
Depends on who you ask. Some would say that the Hildesheimer Rabbinical Seminary in Berlin was the first “modern” MO institution so MO traces it roots to there. Others would claim that Rishonim like who were up secularly educated were haskaficlly MO so they trace their roots to them. Others would claim that RYBS from Boston and YU was the founder of MO. And others would claim…
You would get conflicting claims if you were ask about the current variations of many other movements as well. While no one disputes that the Baal Shem Tov was the founder of Chasidus, many would question how strongly he would identify with the current practices and haskafa of certain Chasdish groups today.
>>> And what are the commonalities that all MO Jews share?
The answer to that would depend on how you define MO. As above one can ask that about many other groups as well
>>>Is a Rabbi ever required to be consulted or is each person able to decide everything on their own?
No learned or deeply committed member of any frum community believes that each person able to decide everything on their own. MO included. Am Haartzim and apathetic people rarely have halacha shaalos. That isn’t unique to the MO world