You’ll copy Stream of Thought for each time you paid its replicate cost, even if it’s countered.

Stream of Thought is still on the stack while you choose cards in your graveyard to shuffle into your library. It can’t shuffle itself into your library.

You choose which cards to shuffle into your library while Stream of Thought is resolving. If you target yourself with Stream of Thought, you may choose any of the four cards that you put into your graveyard or any cards that were there before.

If you replicate Stream of Thought, the copies resolve before the original spell, so they can’t shuffle Stream of Thought into your library.

The copies that replicate creates are created on the stack, so they’re not “cast.” Abilities that trigger when a player casts a spell won’t trigger.