Whether or not the ability triggers depends on if the Avatar has the ability while in the graveyard. If it doesn't (thanks to Yixlid Jailer, for instance), then the ability won't trigger. It doesn't matter whether the Avatar had the ability in its previous zone (it will trigger even if it had been on the battlefield with Humility).
Serra Avatar's power and toughness are constantly updated as your life total changes.
The ability that defines Serra Avatar's power and toughness works in all zones, not just the battlefield.
If Serra Avatar is no longer in your graveyard when its triggered ability resolves, you won't shuffle your library.