Wouldn't evil have already existed on Earth before Adam and Eve?
To answer your first question and the succeeding questions let's decipher it one by one in order of Genesis Creation accounts.
Your first question is if evil existed on earth before Adam & Eve?
Let's look closely on Book of Genesis on creation account before Adam & Eve were made on the sixth day.
During the 1st day up to 5th day of Creation, the earth that is formless and void in Genesis1:1-2 is now created ready for habitation, everything was created before Adam & Eve was created. And God said that "it was GOOD".
[1:1] In the beginning when God created the heavens and the earth,
[1:2] the earth was a formless void and darkness covered the face of the deep,
while a wind from God swept over the face of the waters.
So we can see from your first question that the earth, the Paradise that God created for Adam & Eve was GOOD and not "cursed"
Therefore, the answer to your question if EVIL already existed on earth before Adam & Eve ending on the 5th day of creation was NO. Evil does not yet existed on earth or Paradise, the bible called it Garden of Eden.
Now to answer your succeeding question which is the cause of confusions, let's separate first Lucifer account on falling from Heavenly Realms or casted down to earth.
You mentioned Ezekiel28:12 which is the description of Satan, and you are right the common biblical interpretation of this verse is attributed to Satan.
Let's look for for the definition or who is Satan.
Satan is Lucifer in the Heavenly Realms being cast-out.
Satan,[a] also known as the Devil,[b] is an entity in the Abrahamic religions that seduces humans into sin. In Christianity and Islam, he is usually seen as a fallen angel, or a jinni, who used to possess great piety and beauty, but rebelled against God, who nevertheless allows him temporary power over the fallen world and a host of demons.
We can see from the definition Satan is defined as a being/entity that seduces human to sin. So,when Lucifer was cast down and describes in Exekiel 28:12, his purpose why God allows Lucifer to cast down to earth and enter the Garden of Eden/Paradise is to seduce Adam & Eve. The Wisdom of God allows this, to test the obedience of Adam & Eve for their perfection of virtue and prove their Love according to the teaching of Church Fathers.
Adam was given dominion or to rule over all the created animals and to safekeep the Paradise. The paradise or Garden of Eden is a Holy Place as we can see God's presence was there when He Breathe the Spirit for Adam to become a "living soul" (Genesis2:17) And after that God gave the single commandment of not eating the fruit of the Tree of Knowledge of Good & Evil.
So,when does evil enters the picture or in your question when was the earth cursed.
But by the envy of the devil, death entered the world,
and they who are allied with him experience it.(Wisdom2:24)
Genesis3:14 stated the reason why the earth was "cursed".
So you see, the earth was not yet cursed. It only get "cursed" by God when Adam & Eve committed "disobedience" to God's command not to eat the fruit of the Tree.
"So the Lord God said to the serpent, “Because you have done this,
“Cursed are you above all livestock and all wild animals!
You will crawl on your belly and you will eat dust all the days of your life."
To the woman he said,
“I will make your pains in childbearing very severe;with painful labor you will give birth to children.Your desire will be for your husband,and he will rule over you.”
To Adam he said, “Because you listened to your wife and ate fruit from the tree about which I commanded you, ‘You must not eat from it,’
“Cursed is the ground because of you;through painful toil you will eat food from it all the days of your life.(Genesis3:14-17)
We can see from the punishment God is the one who cursed the ground and the serpent who is indwelt by Satan and Adam & Eve was punished for their disobedience by falling into temptation set by Satan who uses the serpent by indwelling in it.
Lastly, on your question "If Satan did commit evil before Adam and Eve's arrival, then wouldn't the Earth have already been cursed?
Satan committed first evil act of disobedience in the Heavenly Realm and not on the earth when he refuses to serve God's Divine Plan, it's called "NON SERVIAM" or the famous "I WILL not serve", or in the Book of Ezekiel "until iniquites is found in Satan".
Lucifer Fall from Heaven is a separate event from Adam & Eve's Fall which as Genesis narration states Satan indwelling the serpent "beguiled" Eve to eat the fruit, thereby committing the sin of disobedience and Adam follows the disobedience also eating the fruit whom Eve gave to him.
If you are still confused let's uncover the Mystery when was the Angel created?
Definitely, the word angel were never mentioned from Day1 up to Day6.
St.Augustine associated the creation of angel when God said "LET THERE BE LIGHT".
and the passage;in Genesis1:3-4 can be compare in relation to John1:5
[1:3] Then God said, "Let there be light"; and there was light.
And God saw that the light was good; and God separated the light from the darkness.(Genesis1:4)
The light shines in the darkness, and the darkness has not overcome it.(John1:5)
Because this LIGHT is not attributed to the "sun & stars which was created in Day4.
In closing, six days of creation was concluded by GOD saying that "it was GOOD" including the earth that was not yet cursed and Adam & Eve whom was created sinless with original holiness and justice.
Lucifer cast down and take the role of Satan which is an adversary, indwelt the serpent to seduce Eve to fall into temptation that resulted in God's cursing the ground, the serpent and punishing Adam & Eve.