Why does Allah have people born into poverty?
People are born all over the world, so people are also born in poverty. I guess your question is more like: "Why does Allah allow the unbalanced distribution of prosperity we can see around the world?" As I see it most people use outer pleasures to hide their inner discomfort. But the human mind quite quickly adjusts to what it has, so it always needs more and more to push aside the discomfort. This ignorant behaviour is not really working as more and more people (for instance in the USA) get psychological disorders, depressions, drug addictions and so on. As to why Allah allows this? I would say that this is all part of Allah's great plan for humanity. Meaning it is part of evolution. Why would we think about evolution as something that has been and ended? Doesn't it make more sense that it is still going on? More and more people are suffering from their minds/thoughts and emotions and the next step for humanity is to overcome this.
Why does Allah not prevent accidents or disasters from happening and have innocent people die?
If Allah is the greatest, who is in everything and everywhere always, then he is also in every human being acting through all of them. In the end there are no accidents or disasters, just life happening. It is only our judgemental mind that categorizes the outer for us, that sees this as good and that as bad.
Or is it better to not question these things?
Always question everything. Even what I write here.
Also, there are some other things I question such as where people say
when we sleep; Allah takes our soul and during that time we are dead.
But how come we are breathing and there is brain activity even when we
are asleep if we are dead?
There is no such thing as an individual soul. It is just a concept of a higher self that is created by the mind that is afraid of death. So the mind invents a higher self that is going to live on, after the body is no more. There is only life and life alone. All life is Allah the great and only one. When we fall asleep we become unconscious, but as you said the brain activity goes on. When you sleep the mind and your thoughts still cycle around the things that worry you. But other things get mixed in. So our dreams are only the modified versions of our daily though processes. That is why dreams can help you to identify what is on your mind.
Also, what is Islam’s view on Evolution? Islam says Adam and Eve are
the first humans, but then historians say we have ancestors who are
Neanderthals and other non-modern humans. Does that mean Adam and Eve
were Neanderthals or were they modern-humans? What do we believe on
The story of Adam and Eve is only a metaphor. Most stories in the holy books (Bible, Quran, Tao-te-king, and so on) are metaphors that are not meant to be taken literally. Literally most of them don't even make sense. If you ask me, life itself is evolution. It is constantly changing. Allah does not seem to like equal things. No two people are exactly the same. Nor are two leaves on a tree ever exactly the same. Allah seems to like change. So all life is evolution and change.
Is it wrong that I’m questioning these things? Is it permissible in
Islam? Recently I’ve had these thoughts and I’m worried it will weaken
my faith in Islam. Is this a common problem amongst Muslims?
Don't ever allow anybody to tell you not to question something. It's not really you who is questioning. It is Allah acting through you. Question everything always! And please don't believe anything I have written. Maybe just take them as interesting information and something to contemplate about.