From what I know:
Maybe, especially in more "modern" and liberal sects of Judaism. The more liberal sects generally support transgender people, and I should think that it's a small step from support to saying that one can become a rabbi.
As for cantor, I'm not sure.
Yes.
Yes. I personally know trans people who have gone through Reform conversion.
rabbi
a rabbi
The cantor
Rabbi
A Rabbi.
Rabbi
Judaism.
No. The functions of a Rabbi within Judaism are different than the functions of a Minister in Christianity.
A Rabbi or a Cantor (or anyone for that matter) can interpret and teach the laws of Judaism.
Yes, you have to be Jewish to become a Rabbi.
The answer that you're looking for is Judaism, but actually, the person who leads the prayers in Judaism is not required to be an ordained Rabbi. Rather, they need a working familiarity with the prayers, with any relevant melodies, and with relevant Jewish laws. See also:Prayer in Judaism
Judaism is not a centralised religion and does not have a person who's the equivalent to the Pope. The title Rabbi means teacher and today Rabbis have a number of roles in Jewish life.