…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 using one of their Plays on their turn to Play a Keeper from their hand to the table.

You can’t use it to intercept a Keeper being allocated, traded, stolen, or acquired in any other way.

The main four Surprises, of which That’s Mine is one, interrupt or cancel a specific type of card being Played, so you have to ask yourself, “What is the type of card being Played?”

In these cases it’s an Action (or even a Free Action, in the case of Plunder or it’s analogues, in which case no card is being played at all). In those cases all you could do with a Surprise would be to stop the Action itself being played, by using the Stop That! Surprise (aka Avast! Belay That! The Stars Are Wrong!).

Using that, of course, would not gain you a Keeper which is in transit. It would only stop the Action from happening, preventing the Keeper from changing hands in the first place.

*Of course, you can simply 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.