Hey, I'm a Christian, watching your video "Ten Questions", and I would like to answer them as I go along. I'm certain you've received answers before, so I'll try not to be too stringy or extravagant, but straight-to-the-point as I can be. Also, I appreciate the effort put into producing this video, it is very well done.
Question 1 - Why wont God heal amputees?
Answer - Who said He doesn't?
The two facts you stated I must argue on, as I have heard indirect testimonies of healings like that, though they are indeed rare. Jesus replaced the ear on a slave of a high priest (See Luke 22:47-53, incident and context). Many accounts of people being raised from the dead (Biblical and modern).
God is certainly not limited by what we have or don't have, but by our faith- we often limit what we allow Him to do.
Question 2 - Why are there so many starving people in this world?
Answer - God is all knowing, all loving, and all powerful. Again, no prevention on His part, but on ours. And God has told us to "Bare one each others burdens" (Romans 15:1)- What are you doing to feed them?
And as far as "Why good things happen to bad people, and bad things happen to good people", I do know people of either way, but ultimately, those who have given their lives fully to Christ have been blessed, internally as well as externally.
This life is short, and as a Christian, with a hope of an eternity to spend with Him, hardships just seem small in comparison. Though it is not His will that any should perish, and He has sent us.
Question 3 - Why does God demand the death of so many innocent people?
Answer - He doesn't. He wants us to live and to thrive.
God is holy, and no imperfection is allowed or can survive in Him. For any sin, there is punishment- that sin must be removed. Besides, why not just obey God?
You'll see over and over again offering sacrifices in the Old Testament, to "roll forward" there sins, if you will. When Jesus came, He paid the price once and for all, because God is a loving God, and wishes no one should perish. (John 3:16) Jesus told us many times in the New Testament God is relationship, not religion.
Question 4 - Why is there so much anti-scientific nonsense in the Bible?
Answer - There isn't.
The four points you argued are only non-scientific if you consider them from an evolutionistic point of view, one which has neither been confirmed nor proven. But the moment you realize everything in your computer has existed by someone typing words, the moment you realize God really could have spoken the world and universe into existence. I do not wish to argue evolution/creation here, but if you wish to know more, I would be more than happy to tell you whatever you want to know, as well as what may be referenced to this question.
By the way- the Bible has been more scientific then we have been for longer. Look these passages up, you may be surprised.
Isaiah 40:22
Job 26:7
Job 38:16 (First possible to discover in the 1800's with deep-see diving suits)
Ecclesiastes 1:6-7
John 1:3 (Mysterious Strong Nuclear Force?)
As a few examples, though don't look to me for the best of them.
Question 5 - Why is God such a proponent of slavery in the Bible?
Answer - No, that's not what He intended. He intended us to live in utopia, but we sinned. Wherever we are, whatever position, He wants us to do it to our best, and considers no man more than another. Ecclesiastes 7:14 and Ephesians 6:5-9 are some passages that talk about that.
Question 6 - Why do bad things happen to good people?
Answer - Essentially answered in Question 2. But I have a question for you that may lead you to think on this more, which I will ask later on.
Question 7 - Why didn't any of Jesus miracles leave behind any evidence?
Answer - Isn't a person living you knew to be dead evidence enough? Or a lame person now walking clear to see God can do anything? Wouldn't you consider any other document of an eye witness to be valid information? And for the whatever we don't find, it is answered simply because God wants us to believe in Him and His power, not for a robotic computerized response, but out of trust. And He has left behind much evidence so we would believe.
Romans 10:17 is one verse on that.
Question 8 - How do you explain the fact that Jesus has never appeared to you?
Answer - Does He really need to? Jesus, as you reference Him, being still in the flesh- He went to heaven for a little bit to get things ready (John 14:2) and is about to take us back up (John 14:3). He's proven Himself to me, and shown Himself to me His way, and I can't argue with that. I don't deserve what He already has done in me

By the way, the prayer it seems your speaking of asking is one God told us He's not going to answer (Matthew 4:7). But if you truly pray with your heart for Him to prove Himself. He has done it for me and many others, and left us blown away

Question 9 - Why would Jesus want us to eat His flesh and drink His blood?
Answer - I know you understand Christ didn't tell us to eat Him like cannibals. He told us to eat bread and drink wine in remembering Him (1 Corinthians 11:23-26). There are two illustrations you can draw from what He said. Yes, one is graphic- the spilling of His blood for our sins, a reminder of the price He paid- in our place. Dying for our sins in the most horrid way a man could, between the scourging and Crucifixion.
The other, the complete opposite- a beautiful reminder that He is a part of us when we put Him in our lives- He is what moves us, what energizes us, what were made of, and so much a part of us we're fused into one- not a grotesque, gory depiction, but the image of a romance so divine and beautiful between a God and His creation.
(Matthew 26:26-29 for another account)
Question 10 - Why do Christians get divorced at the same rate as non-Christians?
Answer - Really, they don't.
That number, last I heard, was statistically higher. But the problem with that statistic is who the Christians are. By name or lifestyle. Real Christians, who know God and actually know real love, wont get divorced. Because they have realized love is not an emotion, but a choice-
And God doesn't take that choice away from us by forcing us into His will. He created us for love, and there's no point in being loved by a robot. Which is also why sin entered into this world, and why "bad" things happen to people. (1 Corinthians 13 on true love)
So, is any of these reasons sound like convoluted rationalizations, please let me know and I will explain them better, or have someone else do so.
So far I haven't created any strange rationalizations or excuses, have I?
Also, in saying "All scientific evidence supports this conclusion (That is, that answers to prayer are coincidence)" makes perfectly fine sense to me when I realize "coincidences" don't actually exist, but are Gods work instead.
And if indeed I am delusional, the Bible tells me to be a righteous, moral person loving everyone as myself- not a religious, persecuting cult. What harm can be done to society by love? And if your right, what standards do you have to go by to make yourself not?
Now, I have a few important questions to ask of my own, and I ask you to answer them honestly, as a rational, educated person that really does want to know the truth, or at least how I have answered yours.
1 - Without God, can you really define what is "good" and what is "bad"?
2 - Without God, why do you exist, and what hope do you have for anything?
3 - Without God, what is love?
4 - Why should you care?