Tom, God forgives our sins and remembers them no more forever. So God can choose not to know something. Black letter Bible.
If it suits His purpose not to know something, He does not know it. For example, when God stopped Abraham from killing Isaac, He said, "Now I know" indicating that before Abraham had drawn his knife to kill his son, God had chosen not to search Abraham's heart for this piece of information.
No, it is not possible to do something different from what God foreknew you would do. No plan of God can be thwarted, so what He foreknows, He has predestined, and what He predestines He foreknows.
Yes, my view changes the orthodox definition of omniscience, from God knows everything imagainable includeing the future exhaustively, to God knows everything He has chosen to know. Using this definition the doctrine is biblical, using the older one, the doctrine is unbiblical.
No, this change in the defintion of omniscience to agree with the Bible, does not require an change in the characteristic of God being immutable. But again, we must have a common understanding of that doctrine to be sure it has not been altered.
Here is a definition off the internet:
The perfection of God by which He is devoid of all change in essence, attributes, consciousness, will, and promises. No change is possible in God, because all change must be to better or worse, and God is absolute perfection. No cause for change in God exists, either in Himself or outside of Him.
But this does not preclude God changing His mind, and giving mercy rather than justice or giving justice rather than mercy. He can say, if you do this, I will do that, but if you do something else, I will not do that, but will do something else.