1) There really isn't a conversion process in a typical messianic congregation. Its done somewhat like a protestant church where... you go and bingo, you're magically a Messianic.
I'm being a bit light hearted here, but there certainly isn't a conversion process.
However, there are some more orthodox in expression of their Jewish view. These Messianic congregations do practice conversion, but this is the teeniest of minorities.
A Jewish conversion is different between a man and a woman. A man must be circumcised. If he is already circumcised, but not ritually so, then they "prick" the male organ and draw a bit of blood to allow him to enter the covenant. Then he will take a Mikvah, which is similar to a christian baptism. A woman will simply undergo the Mikvah. Also, there is typically a 1-3 year process where you study Judaism with a Rabbi, depending on your knowledge level. You will then stand before the Beit Din (like a council of sorts) and they will ask you why you are converting among a few other questions. After this you then take your mikvah.
But again, conversion is not typical among Messainics.
2) This depends on where you go. In order for one to be considered a "true" Rabbi, they must have a simcha (ordination) from a Yeshivah. A Yeshivah is like a christian seminary but much more intense. Yeshivah usually lasts about 7 years and you study very deeply the religion of Judaism, Hebrew, Aramaic, and many different texts. You would then be a 'student Rabbi' for a year before you receive your simcha.
However, very very few messianic leaders are actually Rabbis. The majority are pastors, etc. and you would typically call them a "congregational leader" or whatever title is used at that congregation.
There are a few Messianic leaders that are Rabbis, but not many.
3) Most call themselves a synagogue. Others, just a congregation. A few do use the word church.
As you are probably starting to see... Messianic Judaism is greatly varied. There is some of every flavor. Some congregations resemble a church and just meet on Saturdays to distinguish themselves from typical christianity. Others will look and feel just like a synagogue, with the added belief in Y'shua as the Messiah.
4) Yes, the Messianic community accepts the NT, we call it either Brit Chadasha (renewed covenant) or Ketuvim Netzarim (writings of the Nazarenes). The latter is used because the original believers were called Nazarenes. The Nazarenes were the first sect of believers, and the term 'christian' or 'christianity' wasn't used until some centuries later.
Most Messianic congregations will actually use the NT as part of the service. Others, while they may not use it in the service, still accept the writings of the NT.
I hope these answers help.
Shalom,
Yafet.
