We haven't seen how the faith militant handles their trials yet (if they even bother with them) but "Trial by Combat" could be a fairly universal tradition in that world.
Perhaps, it's religious nonsense (and specifically tied to the Faith of the Seven) that people are using to validate the absurdity of trial by combat in the first place.
Then again, the whole system overwhelmingly benefits the side that has more money and is a miscarriage of justice very likely determined by the wealthier and more powerful class, using religion as a mere means to sell it to the masses. That's the kind of thing that the sparrows staunchly oppose and would easily see through.
If they don't allow it I do hope we get to hear their dismissal.
We don't know if they were disbanded before because their trials or punishments were especially cruel or if they were just the general fanatical doucheNozzles (imo) they appear to be.
That or the same thing as now, them fucking with the wrong people.
I think they launched their attack when they did because Jon was taking away wildlings they intended to add to their armies.
They had to have realized that one of the White Walkers was killed by now (the one Sam got). Given this, it's possible that they realized that it was due to dragonglass, and that they'd surmise that it came in from the south (even if it actually didn't). Seeing the ships approach from the south they could have gone in to prevent them from being armed with more dragonglass. And it does look like the one Jon fought was searching for it.
The bigger question is why they didn't bother to wipe out Hardhome sooner, since there was basically no risk to them doing so. It's possible that they couldn't travel this far south or close to the sea until it got cold enough, or that the weather has enhanced their magic. Or maybe they were waiting for their forces to be large enough. Maybe there was some special preparation or condition that was needed in order for them to start the initial mass killing of those villagers. It's also possible that they didn't
want to launch a full scale offensive against the wildlings in the first place but do now that they feel threatened.