Thank you for your question, I think you’re talking about a very important issue. But before I answer, please let me say one thing. Unbelieving is not always it is a matter of facts, because there you have some doctor who smokes! They know that produces cancer but they do that anyway. So, in that sense, many don’t believe in Christianity in spite of the proofs. It is also true that out there are many who have sincere doubts and that’s why we need to be ready to answer them with the truth.
So, let’s go for those Bible “contradictions”. I put those quotation marks, not because I want to be unrespectful to your question, but because whenever we find something that appears to be a contradiction, there is for sure a reasonable explanation. Why is that? Because the Bible is the Word of God and, as Jesus said, He is the Truth, there is no contradiction in Him.
About Genesis 4:16 and Jeremiah 23:23-24, what we find in the Bible is that God is everywhere because of His greatness, but there are few times when He decides to show Him to us in a special way. And you can see that in many parts of the Old Testament (with Cain, with Moses in Exodus 40:34-35, with Solomon in 1Kings 8:10, etc.). And also you see that in the New Testament when the Holy Spirit came to dwell in us. He is everywhere, but dwelling in us means we have an especial relationship with Him, a communion.
About Genesis 6:3 and Genesis 9:29, the right interpretation seems to be that God was not talking about the age of every single human being, but He was talking about that specific generation. He will be patient for 120 years since that declaration. You can see some relation between Genesis 5:32 and Genesis 7:6. So we can say that God decided to erase humanity because of their evilness but Noah found grace (Genesis 6:8).
About Genesis 6:4 and Numbers 13:33. In Genesis, we see a mention of “the Nephilim” and it seems to be a reference to giant men rather than a race. But whatever is the case, the reference in Numbers 13:33 came from the fearful spies so it is an exaggeration of what they really saw, because from there on all the reference to those people is “sons of Anak” (Numbers 13:28, Deuteronomy 1:28, Deuteronomy 9:2, Joshua 11:21). The name “Anak” appears in some ancient Egyptian texts referring to a Palestinian governor and probably the people who live there at that time could be his offspring, but for sure they were very tall.
About Genesis 6:6 and on, and Numbers 23:19-20 and on. What we see in the second group of verses is an eternal truth: there is no variation in God, He is the same yesterday, today, and forever. And what we have in the first group of verses is what is called “anthropomorphism”, which means that God receives a human characteristic to help us understand what is happening. For example, when we read in Psalms 119:73 “Your hands” we don’t have to spect that God actually has hands, we know He is spirit, and therefore He has not a human body. As well as that last example, the first group of verses expresses something really complex as the intersection between our prayers and the perfect and eternal will of God but in a human level of interaction.
And about Genesis 6:19-22 and Genesis 7:2-5. What we have there is the same case in Genesis 1 and 2. In both chapters we see God creating humans, but we see different narratives, and that’s why because we are not reading a newspaper of our times, we are reading ancient documents. In our century we are used to reading about someone’s declaration in a newspaper a word by word text, but in ancient documents what we have to spect is not a word by word documentation but a summary of the most important (but of course faithful to what was actually said). So in Genesis 1 and 2, we don’t have contradictions but complementation, as same as Genesis 6:19-22 and Genesis 7:2-5.
I hope my answers help you in some way!