Spur on with Words of Fire

If you gain presence during the turn (after Growth) and uncover the "any" or another element, can you then use it for other powers?
Yes, you can start using elements as soon as you gain them. (p. 14) Similarly, if you are able to play a card mid-turn, you get the effects of its Elements immediately.
(And if some effect were to cover an Element by returning Presence to your track, or destroy a Power card in play, you'd immediately lose access to the elements.)
(Discussion elsewhere) Tags: "Any" element, Elements, Adding Presence, Indomitable Claim, 13 more...

When an effect lets a Spirit play a Power Card outside of the Spirit Phase, does the Spirit immediately resolve that Power?
No, except in the rare case where the effect explicitly says the Spirit also gets to resolve the Power in addition to playing it (e.g. the "Ethereal Conjunction" Event). The vast majority of the time, the new Power Card works like any other: resolvable in the Phase in its "Speed" box when not in the middle of anything else.
Tags: Playing Power Cards, Resolving Powers, Timing, Spur on with Words of Fire, 12 more...
When an effect lets you play a card, does that count against your allowed number of card plays?
No. The number of card plays on your Presence track only affects how many cards you can play as part of the "Play and Pay For Power Cards" action in the Spirit Phase. Cards played any other way (usually in the Fast Phase when instructed to do so by a Power) are in addition to those limits.
(Discussion elsewhere) Tags: Playing Power Cards, Spur on with Words of Fire, Unlock the Gates of Deepest Power, Gift of the Primordial Deeps (Serpent Slumbering Beneath the Island), 14 more...

In a solo game, do you get bonuses on powers like Boon of Vigor that usually only happen if you target another player?
No.
You can target yourself with powers that say 'Another Player', but any effects that are conditional on targeting someone else do not activate. (Rulebook p. 22)
(Discussion elsewhere) Tags: Solo games, Boon of Vigor (River Surges in Sunlight), Gift of Constancy, Elemental Boon, 6 more...

In a Second Wave game, when can players play the Power Cards carried over from the prior wave?
Any Spirit Phase after the game has first entered Stage III.
If the Stage jumps around (e.g., against the Tsardom of Russia), the it does not matter if the game is currently in Stage III, so long as it has been Stage III at some point earlier in the game. (Note though that the early Stage III Card for the Kingdom of Brandenburg-Prussia does not put the game into Stage III.)
Since the Power Cards don't belong to any Spirit that's still in the game, it's impossible to play them any time other than the Spirit Phase, e.g., with "Spur On With Words of Fire" or "Ethereal Conjunction."
Tags: Timing, Second Wave, Tsardom of Russia, Kingdom of Scotland, 4 more...
When playing Ward the Shores, if you play a Power Card not during the Spirit Phase, can you play it face-down to add a warding pattern?
Yes, subject to the "once per turn" limit.
(Discussion elsewhere) Tags: Playing Power Cards, Ward the Shores, Spur on with Words of Fire, Gift of the Primordial Deeps (Serpent Slumbering Beneath the Island), 4 more...