Roman, Roman, Roman... You’re three years-old and already a theologian... you continue to amaze me. (Joe and Ruthann, he might have to skip Sunday School with this one and enroll him straight into some Catholic university...)
But now to the question at hand: “Did the Holy Spirit die on the cross?”
Ruthann, I shall begin with your intuitions because they are quite wonderful. The Trinity has indeed always existed... three Persons, one God, for all eternity... and so it might be said that each Divine Person was “present” prior to the passion, death, and resurrection of Our Lord, Jesus Christ; yet it might also be said that that “presence” was to be dramatically changed after the Son of God was to break into history, for our sake, and become Man. For after the Word of God would take on flesh, God’s relationship with us, and quite profoundly our relationship with Him was to change forever... (and quite for our good, might I add), but this change was only to occur after the Son’s obedience to the Father, through his Sacrifice on the Cross was completed by His Glorification in Heaven.
And this is why when you read in John 16 Christ's words, Christ is telling his disciples that “if I do not go, the Paraclete will not come”; because if he is not Glorified, his Sacrifice would not have been complete... (for what is a Sacrifice if it is not received?)
So don’t get the impression that he is saying that the Holy Spirit is not yet around (or even not yet able to be witnessed on Earth, for recall that he had already been seen at Christ’s Baptism, “in the form of a dove”)... Instead, Christ was saying that if he did not complete his work of atoning for our sins, we as unredeemed sinners would have never been able to share in the intimate life of the Trinity... which is a participation in their intimate Love... whom we call the Holy Spirit.
But now to answer the question a bit more directly... and to do this, we might think of what is meant when we declare that “Jesus died on the cross.” As St. Thomas said (see link), what we declare when we say those words is that Christ died “as man” but not “as God.” For as God, Christ could not have died; it would be impossible for him to have died... God is God, Existence Himself... the Great “I am”... and the Great "I am" could never have been or ever be the Great “I am not”... but in becoming man, the Word of God, through his humanity ,was able to enter death and so make an acceptable offering to the Father for all of Mankind.
So then when we turn to whether or not we could say that the Holy Spirit died on the cross, we have to decidedly say, “no”; for the Holy Spirit, as well as the Father, and even the Son, in his Divinity, cannot be said to pass out of existence.
But please don’t accept this as infallible, for I am far from it...