Can I activate mana abilities even if I don't need the mana?
Yes, you can activate a mana ability even if you do not plan to use the mana. This is often called floating mana. This means that yes, you can use Karametra's Favor to trigger an Inspired ability every turn. Since the Magic 2010 rules update, mana burn does not exist any more, so keeping this mana in your mana pool until the end of the phase has no downside.