Nowhere does Scripture suggest that God cannot co-exist with evil. Clearly he must co-exist with it, otherwise there is either no God or no evil, and Scripture says there are both. What God cannot do is APPROVE of evil or IGNORE evil, because he is by nature holy. E.g. Habakkuk 1:13 “Your eyes are too pure to look on evil; you cannot tolerate wrong. Why then do you tolerate the treacherous? Why are you silent while the wicked swallow up those more righteous than themselves?”
Habakkuk struggles with the fact that though God is infinitely good, he nonetheless allows evil. But God has given choice to mankind, and that choice would not be real if it did not permit the choosing of evil. God will not forever withhold his judgment—hence the unrighteous, those not covered with the blood of Christ, will ultimately go to Hell. Meanwhile, however, God in his mercy postpones the day of his judgment to give every opportunity for repentance. 2 Peter 3:9 says “The Lord is not slow in keeping his promise, as some understand slowness. He is patient with you, not wanting anyone to perish, but everyone to come to repentance.”
God is indeed infinite, everywhere present. As it says in Psalm 139:
Where shall I go from your Spirit?
Or where shall I flee from your presence?
8If I ascend to heaven, you are there!
If I make my bed in Sheol, you are there!
9If I take the wings of the morning
and dwell in the uttermost parts of the sea,
10even there your hand shall lead me,
and your right hand shall hold me.
11If I say, “Surely the darkness shall cover me,
and the light about me be night,”
12even the darkness is not dark to you;
the night is bright as the day,
for darkness is as light with you.
Jeremiah 23:24 says, “Can any hide himself in secret places that I shall not see him? saith the LORD. Do not I fill heaven and earth? saith the LORD.”
Satan, though evil, is portrayed as coming into the presence of God (Job 1). Though we are sinners, God dwells in our world. Now, in Heaven, and on the New Earth, there will be no more sin or evil or suffering (Revelation 21-22). So then, at that point, God will not coexist with evil, for the only place there is evil will be hell.
Now, it could be argued either that God is in hell, punishing the wicked, or that God sovereignly chooses to withdraw from hell, so that he is not present there. If the second view is right, that might appear to limit God, since there would be one place he wouldn’t be. But the all-powerful God could choose to withdraw his presence from one specific place, as an act of an all-powerful God who can sovereignly and selectively withdraw his presence. To say “God could choose not to dwell in hell” would be no more contradictory than to say “God cannot sin” or “God cannot tempt with evil” (which the Bible actually says). Someone could say, “Well, then, God can’t do something and that proves he’s not infinite.” No. God cannot violate his own nature, but this is not a limitation in the ordinary sense, it is simply a distinctive of his nature. God cannot be evil, but that’s not a limitation, it’s simply self-evident that his nature does not permit him to be unlike himself. It’s like the old question, “Can God make a rock so big he cannot lift it?” No matter how you answer the question you appear to limit God.
In fact, God is all-powerful and all-knowing and everywhere present AND he has chosen to allow and therefore co-exist with evil FOR NOW. But He promises sin is a temporary aberration and that he will restore the universe to sinless perfection through the resurrection of mankind and earth itself (as the new earth).
In any case, there is no contradiction whatsoever between God being everywhere present and for the existence of evil.