First, I'd like to introduce myself and say hello to the board. I'll be posting some questions on convert issues and the prayer forum later.
Second- and to the matter at hand, the three questions:
1: Can God utterly destroy Himself; that is, can God make himself not be, in the sense that a created thing can be utterly destroyed?
2: Can God create another God, one greater than He?
5: Can God trklvmrtfx?
The answer to all of these is "no", by my reckoning. As an ignorant I couldn't say, and I welcome correction, but I suspect the orthodox Church would agree with that as well. This is not, of course, a limitation on God at all; it simply means that omnipotence does not mean the possibility of doing absolutely anything.
I will take the second question first- Can God create another God, one greater than He? The answer is "No, but". No, God cannot create another God, but the Father Almighty may certainly beget a Son. Because of the essence of God existing "now and ever and unto the ages of ages" this begetting happens outside of our concept of time, and before all worlds. This begetting brings about not "another" God but two Persons (I'm leaving out the Holy Spirit for the moment because I have only a vague shadow of the Orthodox teaching on the subject) with one Essence.
The Son is not greater than the Father, but "Being in form of God counted it not theft to be equal to God". (philippians 2:6, translation mine)
This leads us to the first question: "Can God utterly destroy Himself; that is, can God make himself not be, in the sense that a created thing can be utterly destroyed"? What He can do is "Empty(or void, or make himself nothing) Himself, taking the form of a slave, being made in likeness of man, and found in fashion as man, humble Himself, becoming obedient unto death, even death on a cross" (philippians 2:7,8). As to whether or not this death means "utter destruction", the answer I believe is no, but no created thing is utterly destroyed upon death. Far from being "utterly destroyed" we believe that Christ went to the souls of the dead, where John the Baptist was already preparing a way, and set the faithful free. If anything, Christianity was a rejection of the doctrine of "utter destruction" favored by certain Greek philosophers and Hellenizing Saducees; and a sound confirmation of belief in the judgment to come, the resurrection of the dead, and the afterlife. Christ's Resurrection from the dead is the very hope our belief is based upon, the triumph of the Incarnate Word, Only Begotten of the Father, over death and utter destruction.
Truth be told, the two questions seem to really be red herrings, which need to be traced to where they diverge from Christian thought. The first question (in the order you listed them) should have been two: "Is death utter destruction?" and "How can God die?" while the second question should have been "How can two(three) Persons exist in one God, and how can one of these Persons acknowledge another as greater?"
The answer to the first question is "No, at least not the death of the flesh." The second and third question are very interesting, and while I have some theories it is nothing I would care to share. The only answers I can give is "I don't know, yet I believe that God did die" and "I don't know, but I believe the Son does His Father's Will"
Question 5: "Can God trklvmrtfx?"
According to certain Charismatics, yes, most certainly. He can blmpfcrng and prngsbghbu as well.
P.S. Just because I hate to leave the hymn unfinished:
"Wherefore, God has highly exalted him, and graced him with a name above every name, that at the name of Jesus, every knee should bow, in heaven, in earth, and under earth, and every tongue confess that Jesus Christ is Lord to the glory of God the Father".
P.P.S. The above statements are by a simple sinner with no credentials whatsoever, and if anything above does not reflect the Orthodox point of view, please, let me know.