Does the First Commandment imply that other gods exist, that God has peers? Why or why not?

No.

It only indicates that other deities are worshiped.

It does not indicate that those other deities exist, or that they are actual deities.

For example: if I say, "Thor is the Norse god of thunder"

no one assumes that I believe that Thor exists and is a real deity.

Everyone realizes that I am using the term "god" in the sense "worshiped as a deity".

Likewise when the Biblical authors refer to deities other than the Judeo-Christian God.

Yes even the Bible acknowledges the idea of many gods.. (The First Commandment 'you shall have no other gods before me; would be nonsensical if there were no other gods). What distinguishes the many pagan gods from the one Creator God is that the pagan gods are created entities or beings, while the creator God is not. Aristotle's discourse on the Prime Mover demonstrates that there can be only one Prime Mover (Creator God) so your likely objection for many creator gods, was answered about 2300 years ago, by a Pagan Greek Philosopher.

No. An examination of the etymology of the words, as well as the balance of Scripture makes it plain that there is but the One God who is the Creator of all that exists. Other gods are the product of men's imagination.

Genesis shows that some of the spirit persons came to earth the ones God created. They wanted to be worshipped so they set up false religions. Which they do today. So in these false religions they have humans worship them. When they were in Egypt they were surrounded by false Gods. Or the worship of spirit persons. So he has no peers or other Gods he is the only creator of spirit persons, and humans.

Yes, that's the implication.

Other gods exist but none are like unto God.

Psalm 86:8 Among the gods there is none like unto thee, O Lord; neither are there any works like unto thy works.

Exodus 8:10 And he said, To morrow. And he said, Be it according to thy word: that thou mayest know that there is none like unto the Lord our God.

Exodus 15:11 Who is like unto thee, O Lord, among the gods? who is like thee, glorious in holiness, fearful in praises, doing wonders?

Deuteronomy 6:12 Then beware lest thou forget the Lord, which brought thee forth out of the land of Egypt, from the house of bondage.

Deuteronomy 8:14 Then thine heart be lifted up, and thou forget the Lord thy God, which brought thee forth out of the land of Egypt, from the house of bondage;

Joshua 24:17 For the Lord our God, he it is that brought us up and our fathers out of the land of Egypt, from the house of bondage, and which did those great signs in our sight, and preserved us in all the way wherein we went, and among all the people through whom we passed:

Ezra 9:9 For we were bondmen; yet our God hath not forsaken us in our bondage, but hath extended mercy unto us in the sight of the kings of Persia, to give us a reviving, to set up the house of our God, and to repair the desolations thereof, and to give us a wall in Judah and in Jerusalem.

Yes.

Mankind has made idols of their gods out of wood, metal and stone and suffered the consequences for it. <t><

There are certainly some gods worth looking at. In ancient Sumeria there was a deity that I find very interesting. This deity was named El. He had other names, El Elyon, El Shaddai, etc... I am pretty sure his names sound familiar. FYI, this is way before Israel. This God had a wife named Asherah (like the Israelites Asherah Pole). He also had children gods named Elohims (You know, thou shalt have no other Elohims before me, the verse you mention). One of those gods was named Baal Zaor (the Lord on High) whom the Yahwists renamed to Baal Zebub (the lord of the flies). You do know the Israelites came from this area where this deity was worshipped, also that Baal and Asherah are cults within their religion? Yet this God is a Polytheistic God who existed alongside many others. Not the Monotheistic deity that eventually emerged.

The Israelites took many stories from the ancient Sumerians. In the epic of Atra Hasis we see the Gods forming men out of clay statues that they then breathed the breath of life into them. This was because we breathed and we knew that was needed for life. They then tried putting trinkets into their Gods mouth (the statues) and then blowing the breathe of life into them( it is called a golem spell). This eventually became where the gods made statues and blew the breathe of life into us. What do we see in Genesis? God formed man from the dust of the ground (his name Adam literally means red clay) and breathed the breath of life into him. Or what about the Noachian flood myth? This came originally from Ziusudra (I think that is how you spell it) a wealthy farmer around 4,900 years ago. The Iraqi flood plains had a several year flood that was about 20 feet total. He took some of his animals onto a boat and saved them. Much later we get in the epic of Atra Hasis another version of this story. The Gods had made men (as described prior) to do the work of the gods, but the men were noisy and made the gods angry. They drowned them in a worldwide flood that was roughly 20 feet deep). Later we get the epic of Gilgamesh who tells a story of Utnapishtam who was spared by the gods while they flooded the whole world (also at a depth of around 20 feet which covered everything that could be seen), he spared all manner of animals, he had one window in the boat that he opened up to release on bird, then another then another, then a dove finally returned back with an olive branch, They then landed and he performed a sacrifice. Noah's flood story was 20 feet above the tallest mountain. Otherwise every major point in this myth was taken from these previous versions. Also, the dimensions of the boat were different.

There are many other stories of the ancient Sumerians that the Israelites adopted as their own. Enough for now though.

The Holy Scriptures say that there are many gods. Satan is the god of this system of things, money can be a god, your belly can be your god, etc.

Other gods would be fallen angels. They are nowhere near in the same league as God.