I fear that the mindset expressed in these BuzzFeed videos is becoming more and more common-- especially by people who claim to be Christians! In other words, a palpable split is being promoted within the Christian Church.
That is no surprise. Scripture has warned us that this would happen. It is grieving, but not surprising. The pastoral concern comes for the humble Christian who is beseiged by these challenging questions. At first, we are often dumbfounded by these challenges, because we have never heard them (and their thinly veiled accusations) before. And since we have no immediate response, we assume that their questions/challenges are valid and that we need to change our beliefs.
Another pastoral concern comes when we strive so hard to be liked and accepted by people who dismiss, reject, and mock parts of the Bible. All people like to be liked. Guess what? Jesus said that the world will hate us because it hates him. Though this is a minor form of persecution, it is just that. Those who hate Jesus will not like us unless we abandon Jesus. We should remember that, and then turn to the Lord all the more when we have to endure such hatred by people who claim their allegedly better and more enlightened Chrisitan faith. The sad fact is that many of these so-called Christians confess a heavily-edited-for-the-culture faith, which is an abandonment of much of God's word. We should recognize this, expose it for what it is, and be prepared for the hatred of those who boast of a greater love than we have.
Rev. Hans Fiene, (LC-MS) has written an excellent article in the Federalist to respond to the BuzzFeed video. You can read it here. It is at least one way Christians can respond to the challenging questions which are being lobbed against them. You can also listen to an interview here with Rev. Fiene on Issues, Etc. (Note: Listen to Issues, Etc. often. They offer outstanding programing, and it is all archived for your listening convenience.)
Another way to respond is to refuse to accept the premise that comes with each challenge. Most challenges may have a grain of truth in them, but do they truly represent the Christian faith? Some Christians might be guilty of the charges, but does that reflect what the Bible actually teaches? That should be considered before you make any response.
Christians, they are challenging your faith. Do not give in to them. Continue to stand on the one, firm foundation which is Jesus Christ and his word. Though they may hate you for it, Christ will love you, strengthen you, protect you, and finally deliver you. Meanwhile, love and pray for those who persecute you. Their future is much more frightening than yours.