Planeswalkers with indestructible will still have loyalty counters removed from them as they are dealt damage. If a planeswalker with indestructible has no loyalty counters, it will still be put into its owner's graveyard, as the rule that does this doesn't destroy the planeswalker.

Boros Charm's second mode affects only permanents you control at the time it resolves. It won't affect permanents that come under your control later in the turn.