Cult of Gigareth

The Cult of Gigareth has a goal of returning Gigareth back to the world. It is believed that Arhor'ha exists on the back of Gigareth, so they are desiring to return the world to the way it should be; an infernal land free from the dominion of others. A land of anarchy and chaos. They desire to reverse the actions of the Great Game that occurred 100 years ago and reunite Gig and Reth. Destroying the enemies that prevent their absolute reign.
Recently, Ty Targarius lead a purge to rid Arhor'ha of the Cult. This resulted in the almost complete destruction of Nexus, as this was the only place they seemed to operate openly.
After the purge and cleansing of the nexus; are the regular people hostile to the cult or do they not care/understand? Regular people are hostile to infernals before the purge. Nexus was a tolerant city-state. Nexus, under elvish occupation, is now extra hostile.
Is there a kill on sight order for the Targs and those loyal to them when they encounter a breaker or builder? No. That is generally the job of the Candles to find the infernal threats and coordinate with the Royal family for the correct response. Individuals would have their own thoughts on the matter out of fear, hatred, or sympathy.

People are allowed to leave as the cult believes in anarchy and forcing people to remain goes against the belief. That said, most people that leave are put in a very dangerous path as other believers or cultists don't think twice about killing them. Leaving is one thing. Having the strength to survive after leaving is another.

Structure

The Cult of Gigareth only operated openly in Nexus. There were priests and they had believers. The way they operated otherwise was more Pagan in nature where they arrived for ceremonies and rituals that the priests coordinated. The same is true for where the cult operated in other city states, but the priests coordinating were required to do so in secret.
A High Priest was capable of interpreting the will of Gigareth and had responsibilities such as finding the Dark Heart Holder for the great game, Builders, Breakers, other priests, and expand influence.
There is a great deal of freedom given to the priest's flock.
Priests themselves are chosen by their ability to listen or feel Gigareth's intent. Its a sort of attunement. High Priests are priests that have demonstrated that ability frequently and go without question that they can sense those with affinity to Gigareth. Its like how all metals are conductive for electricity but some are more conductive than others. The only way to know for sure is to get results.

Culture

They have holidays, however, they are not easily tracked on a calendar as they have to do with celestial positioning that causes phenomenon like red blood moons or specific effects to the aurora in the sky or ley lines on the ground. The two well-known events are 'The Ascent' and 'The Descent'.
Some priests believe that the current state of Gigareth is due to an outcome of The Descent and an equally powerful Ascent Day will balance the effects.