This is a good question. To answer requires a certain level of speculation and reasonable argument, which does not rise to the level of proof. Therefore you should take my answer with a grain of salt. God does not provide clear cut answers to every possible abstract philosophical question in the Bible. Bear in mind that those who raise this question are very often coming from a Calvinistic perspective. They presuppose that God’s soveriegnty trumps our free will. In other words, most Calvinists deny that we have free will in the broad sense I would use the term. They use the argument you raise here to support the theological position of predestination. They try to logically prove that either God changes or we cannot freely choose to be saved (this is an example of what is known as a false dilemma). Based on human logic/common sense, if we commit to the belief that God does not change, they hope to force us to choose the latter conclusion that we are predestined and not free to choose to be saved by our own will.
I believe that God does not change. In fact, this is about as well-established biblically as any theological statement. (Numbers 23:9, 1 Sam 15:29, James 1:17). If this is true, then we can conclude that God did not change subsequent to creation. Obviously his relationship with his creation changed when he created us, but because his relationship with us changed, that does not mean God changed.
This does not make the answer to your question easy, but it simplifies the possible ways to think about it. Bottom line, no, God did not change when we were created. Early Christians who were strongly influenced by Greek philosophy, such as Clement of Alexandria and Origen struggled to explain how Jesus "changed" when he took on human form. Did the Son change somehow in his essential nature when he became flesh, or did he remain the same essence but simply chose to limit himself while he was in the body? I believe the second option is correct, but this is not an easy question.
Getting back to your question, I am not sure what one would mean to say that God "entered" time. What does that mean? I think you are asking whether God lost his timelessness or lost the ability to see all reality at once, but now can only think linearly with respect to time. This would mean he cannot see into the future. If so, my answer is a definite no. God prophesies the future many times in scripture. He is fully cognizant of past present and future. He has not changed. As humans–forced to live in a world limited by linear time–it is very hard for us to conceive how it is that God can give us choice when he "already knows" what we will choose. He foreknows but does not predetermine. This seems illogical from a human perspective, but God is not bound by our own human experience or logic. Because people make the mistake of assuming that God is limited by logic which makes sense from the human point of view, they tend to humanize God, which is a mistake. Now, when Jesus came and lived in a body, it appears that the Son of God DID limit himself, at least in some sense. Perhaps it is true that while in the body Jesus was not fully cognizant of time in the way God is. Of course that raises the possibility that God the Father has also limited himself after we were created. I see zero evidence for this and positive evidence against this, as God is clearly able to prophesy the future. He knows what will happen before it happens. This is consistently taught in the Bible.
For us God is a mystery. I am not sure you will be able to "make sense" of how God who exists outside of time can relate to us who exist within time, but He does! I know that God responds to prayer because scripture says he does (and the Bible is inspired by God) and my experience agrees with this.
As for God changing his mind, this is a difficult thing to comprehend. Yet we have the Bible describing God being influenced by our prayer. It is not that God changes, but that God, because he loves us, responds to us as a parent does to the request of a child. If someone asks me to do something and, out of love and consideration of that person, I change my mind and decide to do that thing, this does not mean that I have changed who I am. So when God "changes his mind" he is not changing in the larger sense. He is exactly the same after he resonds to our pray as he was before we even prayed. Even as human beings we understand that we can change our mind with changing our selves.
If we can agree that God can "change" his mind–in other words can be influenced in his will by what we do or ask without changing in the larger sense, then this seems to come close to answering your question. Even with this, it is hard for us who are limited to thinking of time in a linear way–with past present and future–to conceive how God can know what we will ask and change his mind, when at the same time he already knows what we will ask and that he will change his mind. I will admit that this raises a logical quandry for us, but I have to conclude that for God this is not a problem.
I hope I did not cause more confusion than clarity by my answer.
John Oakes, PhD