1. If predestined, why is there a need for judgment?
First let me address why there is a need for judgment, biblically, and then I’ll address how predestination plays into this. We, the human race, are to be judged because we are sinners on two accounts. First we are sinners because of our actions, but our actions are predicated on who we are by our natures. So secondly, we are sinners because we are a “fallen race” due to the decisions of our parents, namely Adam and Eve. Reformed Theology teaches that Adam and Eve had “free will” but we do not. The Westminster Confession of Faith says that “Man, in his state of innocency [Adam and Eve], had freedom and power to will and to do that which is good and well-pleasing to God.” In other words Adam was able to sin (which he did) but also was able to not sin (which he didn’t). Now, because Adam chose to sin, he and his descendants (us) “wholly lost all ability of will to any spiritual good accompanying salvation…dead in sin.” So all of humanity is fallen, broken, in slavery to sin, and dead in sin. We are all deserving of judgment because of the “Fall” and because we also commit wrongs daily, unable to not sin. God chooses, or predestines, some to life. So the doctrine of predestination is that out of this fallen race, God chooses to save some. God, because He is all-knowing and stands outside of time, knew who he would choose before the foundations of the world, and also knew that he would choose us not because we chose him (how can we? We are dead in our sin) but based on his own desire / good pleasure. This is the historical understanding of predestination.
Now, if you are asking why God planned everything the way he did, that is a mystery. In other words, surely God knew that Adam would choose to sin, so why did he allow or permit Adam to do that and thus condemn the whole human race. Why allow suffering in the first place? (By the way Reformed Theology would say that God permits or allows evil, but is not the Creator or source of evil). Or why doesn’t God redeem every last person? These are certainly beyond me. Yet God has been faithful, true and good throughout history and in my own life experience, so even though I don’t fully comprehend His purposes and complete plan (As a finite child compared to an infinite God, how could I?), I fully trust Him.
2. A person, not pre-elected, could be a believer and follower of Christ, but not eligible for salvation?
Only a person chosen by God, regenerated by the Holy Spirit could be a true believer and follower of Christ. Those who God does not work in, do not have the ability to heal their own blindness and believe. Jesus indicates that there are those who will never understand, and the truth is purposefully hid from them. Mark 4:11 “And he said to them, "To you has been given the secret of the kingdom of God, but for those outside everything is in parables, 12 so that "they may indeed see but not perceive, and may indeed hear but not understand, lest they should turn and be forgiven.” So the case which is presented in the question cannot occur, because you must be elected, chosen by God, by grace (a free gift), in order to first believe.