Storm: Storm is a triggered ability that functions while the spell is on the stack. "Storm" means "When you play this spell, put a copy of it onto the stack for each other spell that was played before it this turn. If the spell has any targets, you may choose new targets for any number of the copies." See rule 502.30, "Storm."
Mind's Desire {4UU} |Sorcery| Shuffle your library. Then remove the top card of your library from the game. Until end of turn, you may play that card without paying its mana cost. (If it has X in its mana cost, X is 0.) / Storm (When you play this spell, copy it for each spell played before it this turn.)
Storm creates copies of the spell. They can be countered like other spells.
The ability triggers upon playing the spell and not upon resolution. Countering the original spell only counters the spell, not the storm. When each copy resolves, you do the action it tells you to do, which is shuffle your library and remove the top card of your library from the game. You get priority between each resolution of the copy to play spells.
If you don't play the spells, they stay removed from the game, as the spell doesn't say anything about doing something else with them.
Note: You can shortcut "shuffle multiple times and reveal" with "shuffle once, reveal until either of you wants to play something" for the sake of time.
