We can answer these questions by analyzing Moshe’s prayer. He mentioned the promises made to the patriarchs. Now, who was greater, the patriarchs or Moshe? From the earlier parshios in Sefer Shemos, it would appear that the patriarchs were greater. Moshe mentions them in his prayers, Hashem chides Moshe that the patriarchs never questioned Hashem, even though they never saw the fulfillment of the promise to make them into a great nation and give them Eretz Yisrael. Chazal say that only three of our ancestors are referred to as patriarchs. For example, we only mention Avraham, Yitzchak and Yaakov in Shemoneh Esrei, excluding Moshe, On the other hand, we say that Moshe was the greatest prophet, including all prophets who preceded and succeeded him, including the patriarchs.
Let us compare Avraham and Moshe more closely. Avraham sought Hashem on his own, without receiving any help from Hashem in his quest. The Rambam (Hilchos Avoda Zara, Chapter 1) says that the world was cascading down hill until the pillar of the world, Avraham, was born. At a young age, either 3 or 40, he began to question what controls the world. Even though he had no teacher to guide him and he lived in a family and community of idolaters, his fantastic intellect led him to conclude that there is a God that controls everything.
The earliest recorded conversation between Hashem and Avraham was when Avraham was 75 years old. In the intervening years he debated the idolaters and defended his belief in the one true God. During these years he was mocked by many. He must have been asked many times “Avraham, you believe that there is a single God? Have you ever spoken with Him? Has he communicated in any way with you?” Yet he dedicated his life to spreading his monotheistic belief system, one person at a time. He received no assistance in this task from Hashem. The Torah does not relate any supernatural miracles (in comparison with the miracles that occurred to Moshe and bnei Yisrael) that were performed on behalf of the patriarchs. Even the miracle of Sarah bearing a child at the age of 90 was not as blatantly remarkable a miracle as the splitting of the Red Sea. Yitzchak and Yaakov were not the beneficiaries of obvious, supernatural miracles either, even though events recorded about them suggest the guiding, hidden hand of Hashem. The patriarchs attained their special status because they popularized the name of Hashem without His help, and they dedicated their lives to this task, despite all the hardships they endured.
The term Elokei Avraham (G-d of Avraham) is in the possessive form. By Avraham searching for and finding Hashem, who was rejected and discarded by the world around him, so to speak, Hashem became the possession of Avraham, so to speak – similar to one who claims an ownerless object. The possessive form is used for each of the patriarchs. In their own way, each patriarch searched for and found Hashem, taking ownership of Hashem in his own unique way.
Moshe came to learn of Hashem in a completely different way. Hashem sought Moshe at the burning bush, not the other way around. Moshe debated with Hashem for a week in an attempt to convince Hashem to send someone else to redeem the people. When Moshe finally agreed to go, he was given a scripted set of messages and signs to deliver to the people and plagues to visit on Paroh and the Egyptians, to convince the people and Paroh that their redemption had arrived. At the time of the Exodus, Moshe did not win the right to be the leader of the people. Rather, Hashem gave it to Moshe because the situation demanded it. Moshe did not have to go through an akeidah like Avraham did in order to attain his leadership.