No, God cannot do everything. God is holy and He cannot sin. The Bible tells us He cannot lie (Hebrews 6:18; Titus 1:2). Also, since God is eternal by nature (Psalm 90:2), He cannot stop being God. He cannot deny Himself (2 Tim. 2:13). God cannot be tempted by evil, nor does He tempt anyone with evil (James 1:13).
Some critics of Christianity will say that if God is so powerful and He can do anything then He should be able to make a rock so big He can’t pick it up. If He could do that, then there would be something He could not do. But such absurd and illogical challenges are meaningless. The truth is that God must be consistent with His own nature, and He cannot violate His own nature. Therefore, God cannot lie, cannot stop being God, cannot deny Himself, and cannot be tempted by evil. Why? Because He is God.
The thought that God could sin violates all we are told about God in the Bible. Unless one reduces Scripture as we have it today to fallible and merely human transmission, without any Divine oversight/influence, I cannot see how any genuine Christian could even think that thought, much less express it publically, as I have actually heard of late.