With your analysis you can also argue that love doesn't exist either.
No you cant, because, as I said, God is Love
, therefore Love is Eternal, without beginning, without end, Uncreated. God is a Trinity, and each Person of the Trinity Exists in a Loving Community with the Other Persons of the Trinity- The Father Loves the Son and the Spirit, the Son Loves the Father and the Spirit and the Spirit Loves the Father and the Son, and there was never a time when this Loving relationship did not exist, therefore Love is Uncreated, God is Love and Love is God. Love exists because God is Love and God Exists.
If, as you suggest, evil is "uncreated", that means that evil is equal to God, and just as Almighty as He is, and the battle between good and evil is equally pitched and could go either way. This is the teaching of Zurvanism
, a branch of Zoroastrianism which holds that there are two equal and opposite eternal gods, Ahura Mazda (the good one) and Angra Mainyu (the evil one).
Our actions reveal who we are. Hence we are evil or good.
No. Our actions reveal what our actions are, not what we are. We are created in the Image and Likeness of God, and our inherent nature is Good- God Himself has proclaimed this: "And God saw everything that He had made, and behold, it was very good."
Who we are can never be effaced.