New information about Cindered Shadows is published regularly on the official Japanese Twitter account of Fire Emblem Three Houses. They complete what we had already summarized about the expansion.
The expansion is played separately from the story, from the title screen, and with its own saves. We don't know if NG+ bonuses will be applied, but completing this secondary story will have an impact on the main story. By resuming your main game, you will be able to talk to an NPC in the monastery to visit the Abyss, which should give you access to its features.
Details on the timing and conditions for the recruitment of the 4 new students, Yuri, Balthus, Constance and Hapi are yet to be announced, but we know that you will be able to involve them in the various basic activities, such as tea parties, among others.