”Day as we know it, begins with the morning and ends with the evening. Why does the evening come before the morning at the end of each of the six days that God worked in Creation in Genesis chapter one?”
The principle question is why must you assume the evening is the end of the day? Says who? Modern society? Sure that’s how we are forced to set our clocks if we want to partake in the marketplace.
If God says evening, going into the night comes first, then that’s the beginning of the day and morning going into the daytime, is the end of the day.
It’s backwards to say that since the Hebrew people considered the day started in the evening that this is why Genesis 1 reads as it does. Rather because God said the day starts with the evening that is why the Hebrew people started their day in the evening.
In light of this your subsequent question
”Could it be that God created evening (the end of day) before morning (the beginning of day) because He is unique in that He is the only being in existence who has no beginning?”
becomes moot as you are contrasting two entirely different paradigms and the axioms contained within them, are by nature going to conflict.
Possibly you meant to ask why does God consider the darkness to be the first part of the day, the beginning and the day to end with illumination, with daytime? How does this reflect on His character?
In order of events, He created darkness first
“I form light and create darkness; I make well-being and create calamity; I am the Lord, who does all these things.”
Darkness was over the cosmos, heavens and earth, or over the waters.
“The earth was without form and void, and darkness was over the face of the deep. And the Spirit of God was hovering over the face of the waters.
Afterwards God brought forth the light and had to separate the light from the darkness.
And God saw that the light was good. And God separated the light from the darkness.”
Your question implies that darkness is uncreated. That doesn’t correspond to the text, even if it agrees with modern western thinking. Darkness cannot exist unless first there is somewhere for it to exist in. Prior to the beginning, prior to the creation of the cosmos there was nothing, meaning darkness itself did not exist either.
As such evening being first is an acknowledgment of chronology. The first day, which started in Genesis 1:1 at the beginning, at 00:00:00, corresponded to the evening time on subsequent days. When the sun went beyond the horizon around its circuit that was the end of one day and the beginning of the next. (And the sun was only placed in the heavens on day four, it didn’t exist prior to day four).
If we want to understand biblical prophecy and timing is of the essence then it pays to know when the day starts.
“But at midnight there was a cry, ‘Here is the bridegroom! Come out to meet him.’”
If you think with a modern mind you would think midnight is the end of the current day, when in fact it’s already hours into the next day or what we would call tomorrow.
This passage if it has any prophetic correlation, and you were to understand it thinking we are already at the seventh day, the calendar is going into the eight day already, something should have happened already, you’d be misled because midnight is in the eight day and the Calendar would be expected to flow over into the eight day (thousandth year).