"In 3 Nephi 28, the 9 disciples are promised that when they die they will go to "my Kingdom" meaning Christ's. However, the other 3 who tarry are promised to go to the "Kingdom of my Father." Are they different? They must be, but how? In what way? Different levels of Exaltation? This same thing is discussed in D&C 7. Peter is promised "My Kingdom" while John is promised the greater blessing. I'm assuming it's "my Father's Kingdom" like the 3 Nephite disciples."
I've written about this in Beloved Enos. The offer is extended to all those whose calling and election is confirmed. The 9 chose to move into the post-mortal inheritance at their death. That is, they would not be required to return here for anything else, but would be judged, crowned and exalted upon death. Because this is a blessing conferred by the Son, it is "His Kingdom" into which they will move. When the work is at last completed and delivered to the Father --at the end of the earth's temporal existence-- it becomes the Father's at that point. The 3 will be awaiting that moment to receive that inheritance. The 9 will enter into the "Son's" until then, and will likewise be among those who are received by the Father, in the due order of things.
[My answer provoked a follow up question:]
"But doesn't Peter, James and John have the earthly role of teaching Adam and Eve (us) further light and knowledge as shown in the temple? Do they send ministering angels or maybe even John since Peter and James don't come to earth anymore?"
Peter, James, and John were added to the endowment by Brigham Young, but weren't part of what Joseph originally portrayed. They were added to remove required narration. When added, they are a "type," and not intended to be the personalities or individuals. Much like Elias is a "title" and not a name. Peter, James, and John are in the endowment types, or "titles" - not intended to be the actual persons who were known by those names while in mortality.
The endowment used to include the words, "You should consider yourselves respectively as if Adam and Eve. ...This is simply figurative so far as the man and woman are concerned." The same could be said about other roles - which all represent truths, but the truths are not tied to personal identities. You are Adam. The endowment is about your life. Those true ministers who are sent are explained in D&C 130: 5, which include those who do (i.e. currently living individuals who have gained a message from the Father and Son to be delivered) or have (i.e., those who have left mortality and are returning as angelic, or resurrected, or translated individuals, who have gained a message from the Father and Son to be delivered) belonged to this earth.