A planeswalker with indestructible still loses loyalty as it's dealt damage. It is put into its owner's graveyard if its loyalty becomes 0.

Because damage remains marked on a creature until the damage is removed as the turn ends, nonlethal damage dealt to a creature you control may become lethal if Avacyn leaves the battlefield during that turn.

A battle with indestructible still loses defense counters as it's dealt damage. If it's a Siege, it will still be exiled when the last defense counter is removed from it, and its controller may still cast it transformed without paying its mana cost.