Did Shadow the Hedgehog die

No, Shadow is ageless.

But if you are refering to Sonic X when everyone assumes that Shadow died after defeating Bio Lizard (Final Hazard) Then the answer is still no.

He appears later in Sonic Heros when Rouge finds him in a stasis tube, so he apparantly survived the fall back to Earth and Dr. Eggman (Or as some people know him, Dr. Robotnik) found him and put him in that stasis tube and cloned him (dicovered in later gameplay in Sonic Heros)