אדון עולם
It is possible to explain why the morning prayers begin with this prayer.
The Gemara (B’rachos 7b) states that from the day the world was created no one called to HaShem in the title of אדון until Avram (Bereshis 15, 8). The three Avos established the three daily prayers. And Avrohom as the first of the Avos established the first prayer, Shacharis. Therefore, we begin the prayers with a prayer that references Avrohom in recognition of his initialization of the prayer services.
מגן אברהם
HaShem also protected Yitzchak (at the Akedah and from Avimelech) and Yaacov (from Lavan, from Esav and from the people of Shechem). Nevertheless, this modifier was applied specifically to Avrohom since the verse אנכי מגן לך (בראשית טו, א) specifically applies it to him.
יראי ה' בטחו בה'
Since other segments refer to Yisrael and Aharon, who is the group “those that fear HaShem”? Thus, this group refers to righteous converts.
Even more, righteous converts are beloved to HaShem from all the multitudes that were at Har Sinai (see Tanchuma on Parshas Lech Lecha). All the people at Har Sinai witnessed the thunder, lightning and the sound of the shofar, yet most of them did not fully accept the Kingship of Heaven. But a convert who saw none of these miracles, did accept the yoke of Heaven, and thus, is beloved to HaShem.
With this understanding we can now understand a difficulty in the Shira, זה א-לי ואנוהו אלקי אבי וארוממנהו. Since the second part of the phrase mentions the G-d of my father, then the first part must refer to a group that does not have such lineage. Thus, the first part of the phrase (זה א-לי) also refers to a convert, who cannot refer to the G-d of his biological father. And due to the elevated status of converts to G-d, the section said by a convert is placed first.