…For example, if they’re using Steal A Keeper, or Exchange Keepers, can I get the Keeper they’re acquiring?
Or if they’re Plundering a Keeper, can I take the Keeper they’re Plundering?
Or if someone plays Mix It All Up, or Share the Wealth, can I get a Keeper that’s being dealt out to someone else?

A: No, No, and No. You can only use the out-of-turn* function of That’s Mine when someone else is putting a Keeper into play in a situation where it was previously NOT in play. You can’t use it to intercept a Keeper being allocated, traded, stolen, or acquired in any other way.

In most cases it’s an Action, or possibly a Free Action, for which there are Surprises you could use – but they wouldn’t gain you the Keeper which is in transit. They would only stop the Action or Free Action from happening, preventing the Keeper from changing hands in the first place.

There are, however, some situations besides normal playing of a Keeper from someone’s hand, where wording on another card does actually include the word “play” in the way a Keeper is acquired. See this FAQ about cards played via the Free Action Wormhole (AKA Mystery Play). Or this FAQ about Actions which allow you to take a Keeper out of the discard pile and play it.

* Of course you could use your That’s Mine card on your next turn, for it’s in-turn function which is essentially the same as Steal a Keeper, so you’re really not in a bad place, even if you couldn’t get that Keeper in the middle of the results of an Action.