Master Alchemist's ability says to pay a fate to the Fire ring, but does not specify it has to be unclaimed. If you are attacking fire ring and activate the ability, do you immediately claim the fate again?

If someone has already claimed the ring, that fate would remain on the claimed ring until someone declares a new fire conflict, right?

The Fate is just placed on the ring and stays there until further notice. The "grab-Fate-off-Ring" part only happens during conflict declaration (barring specifically called-out exceptions like Know the World). Stone of Sorrows can create similar situations with contested and claimed Rings with Fate on them.

With Elemental Fury, I was under the impression that when the ring changed, the attacker claimed any fate on the new ring. Was I mistaken there as well? I could get behind "Attacker takes fate" as being a discrete action as part of declaring the conflict, but I thought there was more of a persistent effect going on for it.

From the rules reference:

When a ring becomes the contested ring in a conflict, move

all fate on that ring to the attacking player’s fate pool.

Basically, claiming fate by the attacker automatically happens anytime the ring BECOMES the contested ring but not WHILE it already is the contested ring.

THAT makes sense, thank you both!