A 72:
One of the conditions under which a creature must roll a save is starting their turn in the area, and failing such a save has a described effect, but there is no effect (such as immunity to future saves) given for a successful save. Therefore, yes, a creature must save repeatedly if they stay in the area.

A 73:
Initiative rolls and dispel checks are both kinds of ability checks, which is one of the categories of rolls on which Foresight grants advantage (it also grants it on attack rolls and saving throws, which between them cover almost all d20 rolls in the game).