Quick Answers to Problems of Evil and Suffering

I'm responding to these assuming you're not familiar with some of the things I'm saying, so apologies if it seems patronising. I'm also going to try and be brief, so there might be some skimmed details.

If God is good, why is there Hell? Infinite punishment for a finite crime.

I imagine you have a particular something in mind when you say 'Hell'. The movies tend to use the medieval imagery of fire and brimstone with goat legged devils with pitchforks tormenting sinners day and night.
That's just not what the Bible teaches.
I personally am an annihilationist, which means I believe that when the time comes, God will bring the saved to the new Heaven and Earth, and the unsaved will cease to exist - much like how atheists think they will end up anyway.
But I think the most common belief among Christians is that Hell is a prison locked from the inside. God gives everyone a choice of whether or not to be with him and follow him. It's not a case of follow and love me or be punished, it's follow me or do what you want without me.
But the outcome of that is by your own free choice, you will have to live somewhere that God has closed off to himself. And that place will naturally be really awful, seeing as God is the source of goodness. He doesn't want you to go there, but if you choose to go there, he won't stop you.

If God is good, why does he let people die and suffer? And all these disasters happen?

A Rough Guide To God's Plan

I would never suggest that I know the mind of God, and I'm wary that there could be mistakes in what I'm putting here, but I find that when discussing various issues, as a Christian I'm coming from a certain understanding that someone else might not have. I might be taking for granted some thing or another that they haven't even thought about. So while I think I'm talking about something obvious, the other person might not have a clue how I can square what I'm saying with anything. Like, I might be talking about the way Jesus suffered for us, while their image of God is the benevolent comfort blanket who makes sure everyone is happy all of the time. The two things don't make sense.

Things like the 'Problem of Evil' and 'Divine Hiddenness' are massive issues for sceptics, but for Christians, they are nothing. They are expected! Somehow, even though these things are necessary in Christianity, some people think they are reasons that Christianity isn't true. It's clearly a lack of understanding about who God is, what we wants, and his ultimate plan.

So, here I want to attempt an outline of what God is trying to do according to Christianity.
It will be an attempt, and I don't for one second pretend to know every detail. The old phrase "God works in mysterious ways" is not helpful to anyone, but he has revealed to us what we need to know on a need-to-know basis. We don't have all of the information but that should be enough.

Anyway Lesson 1: God is not a comfort blanket who answers every prayer with the goal of making everyone on Earth as happy as possible.

Can't God Just Rearrange The Stars And Write His Name To Prove He Exists?

It's often heard from sceptics that they would change their minds and believe that God exists if he does something big and grand like that. He won't do it...

"The Pharisees came and began to argue with him, seeking from him a sign from heaven to test him. And he sighed deeply in his spirit and said, “Why does this generation seek a sign? Truly, I say to you, no sign will be given to this generation.” And he left them, got into the boat again, and went to the other side" -- Mark 8:11–13 

God doesn't just want you to know he exists. He wants a relationship. Showing off his power while demanding your obedience isn't the way to go about that and he knows it.