However, both Christians and Jews must agree, I believe, that G-d Himself did not, in fact could not, have actually died on the Cross, for if this is so, then He would not be G-d. Or am I misinterpreting Christian teaching
You are quite correct, but (orthodox) Christians add an idea that sources in our understanding of the trinitarian nature of God.
After three hours on the cross, Christ, the Son of the Father, and one of the three “persons” of God (a modern synonym might be “aspect,” but this word has a few problems) actually died.
There was, interestingly, a very early heresy that claimed the death of Christ was something like an illusion - an appearance but not an actual death in the human sense. A variation on the heresy taught that Christ’s body was an emanation, and not substantial like ordinary mortals. These theories were expressly condemned, and I am sorry that I cannot recall the name of it for you.
This is not the end of the story under orthodox theology, however. The Father did not (and as you rightly say, cannot) perish, nor does the Holy Ghost (also called the Paraclete) perish. But the human body of the Divine Person - Christ- did suffer actual death, and like (almost) all the souls prior to the incarnation, descended into Hell.
Christ’s body was resurrected, and after 40 days we believe Christ ascended, bodilly, into Heaven, to be reunited with the Father (the formula is “He sits at the right hand of the Father”).
If we think about it, it poses no more difficulty to say “The Son of God died” that it does to say “The Son of God was born.” So far as we observe, whatever is born, dies. The greater theological problem is why, or how, the Father, who is completely outside of nature, could become incarnate. “Blessed be the Lord our God, King of the Universe.” Does the king mingle with the peasants?
Personally, I resolve the the theological conundrum through the understanding of the “sacrifice.” I see the sacrifice of Abraham as a parallel. From Genesis, we know of the more or less total depravity to which humanity had sunk. God wished to raise a holy people, and made a covenant with Abraham. But, there had to be a sacrifice to cover the sin. What could possibly be sufficient for such an incredible, unmerited gift? Only one thing: the people themselves, represented by the first born among them, Isaac.
In His mercy, the Lord commuted the fulfillment of that sacrifice. In orthodox understanding, while comprising the holiest covenant on earth, and creating the holiest people on earth, the Old Law was not sufficient to redeem fully either the Jewish people, or the gentiles at all. A second sacrifice was required, and this the Father provided, just as He had provided the sacrifice for Abraham, who by himself, had no suitable offering. Sarah only bore Isaac because of a miracle.
The sacrifice provided was Jesus Christ, the eternal son of the Father.
So, the sacrifice of Christ was absolutely necessary. But, as some of the posts have observed, it was also an act of love. Creation itself is an act of love, but if we want to examine the “whys” of certain divine acts, I find the concept of sacrifice more helpful.