Once the first ability resolves, the damage prevention continues to apply even if that permanent changes controllers.

Kiora’s second ability allows you to play an additional land during your main phase. Doing so follows the normal timing rules for playing lands. In particular, you won’t play a land as that ability resolves. The ability will fully resolve (and you’ll draw a card, perhaps a land you’ll play later) first.

If Kiora leaves the battlefield after her second ability is activated but before it resolves, you’ll still be able to play an additional land after the ability resolves.

The effect of the second ability is cumulative with other effects that let you play additional lands, such as the one from Rites of Flourishing.

Kiora’s first ability can target any permanent an opponent controls, not just one that can deal or be dealt damage.