The gallery is the quest log. It gives you hints and tells you where to go, if there's no more greyed out characters in the gallery (except for the alternative scene like for Yui) it means you are done for the update.
As for when they are inside their houses, they are meant to have repeatable scenes. They don't have them right now (except for the scene you unlock with the Patreon costumes), but I'll add them in the future.