Making contact with the Jedi Council is, due to the nature of your mission, an extended and difficult process. You DO have the means of getting messages out, down to various encoding procedures, but messages back have to be smuggled into you, so will take time. Actual conversations are impossible, sop frame any questions you want to ask in advance (and make sure Gallagher asks any questions he wants as well as you can't keep sending such messages).
As for the rest...
Synod is an absolutely real world term- it means "A council or an assembly of church officials or churches; an ecclesiastical council. "
You find references to Deel Twain all over the damn place- mostly in stories of corporate acquisitions, takeovers etc. he wasn't lying or being very obtuse when he said that business is his business- he owns more businesses than you can comfortably read in an afternoon. He is a multi-trillioniare, one of the five richest men in thge Galaxy, the absolute richest outside of a known commercial organisation like the Trade Federation, and seems to spend his life shifting stock or making new acquisitions. Whoa!
Father Varm himself has no direct references that you can find.
The Synod of Ferradyne is the ultimate overlaying structure over a complicated set of Churches (that long ago effectively became a single Church) who are dedicated to the worship of the Force. Although its current incarnation dates back 'only' as far as the current version of the Repiblic- 1000 years- its roots go back way further than that. In fact, it would be more accurate to say that the current Synod is a thousand years old, founded after the Sith exterminated the one that existed before (for no good purpose other than that it amused them), but the Church itself probably dates back to the first Republic, right from the very start of things.
The Synod itself is based on the world of Mandragos, a core Republican world that has definitely been there since square one. From there, it is led by the Archimandrite, a figure whose presence in the Star Wars Galaxy is not unlike that of the Pope in the real world (not necessarily in attitude, but in relative importance). The significance of the name 'Ferradyne' is lost but might refer to its founder. The Abbots are the direct servants of the Archimandrite, and spread his word and his policies throughout the Galaxy.
Whilst not the largest religion in the Galaxy, being instead about the third or fourth- probably because it is not a fanaticial religion, and entire parts of the Galaxy are so fanatic- it is probably the most significant in terms of its ties to the Republic, and also the most widespread; there is no prejudice or prequisite amongst its adherents. it is a very safe religion, and so many Senators claim sympathy- few or none of the officials of the Synod have ever run for office themselves, however- they have a Jedi-like attitude to trying not to become actively politically involved.
The Synod are firm believers in the idea of the Force in its natural state- their religious talk is that of the presence of symbiosis, and the Force's role in guiding people and in its role in the creation of all things, including all life. This being the case, the Synod are very interested in the Balance of the Force, and hence champion moral causes and campaign against immoral ones, immorality being akin to evil, which causes imbalance. They frequently preach against the growing corruption of the Senate and try to support Galactic-wide charitable causes, several of which the Synod directly runs.
Not being involved in anything as active as law enforcement, the Synod are total pacifists, speaking out against all aggresion, even under provocation. When the Sith annhilated the officials of the last Synod, they offered no resistance and each, from Archimandrite to Abbot to Priest, was beheaded where he stood. This sparked a popular uprising against the Sith in several areas that they never quite recovered from- though it is ironic that the Synod would not have approved (though they most likely would have forgiven- the Synod is easy going and does not excommunicate for fighting under provocation; they just say it's simply not, ultimately, the right thing to do). A number of worlds still have a martyrdom complex about the Synod since the Sith massacre and ferociously defend its interests rather more than the Synod itself does.
All that is required to be a member of the Synod is commitment to balance (as GL sees Balance as a power of morality and goodness). This theory extends to all things, so an adherent is meant to keep his bodily systems well-balanced as much as his moral virtues.
Nearly all of this is very similar to Jedi thought indeed. But, beyond the total pacifism being a difference, it is notable that there has always been some 'distance' between the Order and the Synod, and they have rarely worked together. They do not entirely get on.