Cards Exhausted through effects other than their own are not saved. Ethereal cards are not typically Exhausted when played, so they will not be saved. Similarly, Deus Ex Machina is Unplayable, and its effect does not count as a play.
Cards played by Havoc and Omniscience are temporily given the "Exhaust" effect, so they can be saved by Strange Spoon after play. If the card is Unplayable, however (e.g. Tactician), it will Exhaust without being played, preventing Strange Spoon from triggering.
As of version 2.0, Strange Spoon interacts oddly with Omniscience. If the card played by Omniscience is saved by Strange Spoon, it will have the Exhaust effect for the rest of battle, and will try to Exhaust every time it is played. Havoc does not have the same issue.
Because Apparitions normally exhaust when played and are incredibly powerful, it may be worthwhile to invest in buying a Strange Spoon if you have them, or in taking the Apparitions if you have Strange Spoon already. Doing so will likely strengthen your deck greatly.
This relic's flavor text references a popular stage magic called spoon bending.
The practice itself was popularized by the magician Uri Geller, and has been tied to psychokinetic or paranormal ability. It has been referenced in popular culture many time, such as a scene in The Matrix, when a young boy bends a spoon to demonstrate the unreality of a computer simulation, and an evolutionary line of a Psychic-type PokémonAbra, namely Kadabra and Alakazam, which are said to utilize spoons to amplify their psychokinetic capabilities.