I believe in God, but several questions bother me regarding His existence:
1. Why is humanity confined to such a tiny portion of the universe, and a small fragment of the universe's time?
2. I see how human acts of evil can be reconciled with existence of God, because of free will, but why is do evil things like earthquakes and childhood cancer arise?
3. Why does increasing intelligence correlate with increasing development of atheism?
Your third question is the easiest, because the premise is mistaken. There is no direct correlation between intelligence and atheism, although there are some spurious correlations. The intellectual culture of our own time is in effect atheist -– it doesn’t deny that there is a God, but proceeds as though there isn’t -- and if you select your data just from this period, you may seem to find a correlation between intelligence and atheism. The intellectual culture of the middle ages was Christian, and if you selected your data just from that period, you would seem to find a correlation between intelligence and Christian faith.
Very quickly, with no discussion, I might add four more brief points. (1) In my opinion, philosophically speaking, the arguments for the reality of God knock the stuffing out of the arguments against. (2) This makes less and less difference to most people, because the intellectual culture of our time is rapidly losing its aspiration to the truth, and therefore its claim to be considered an intellectual culture at all. (3) Although intellectuals are supposed to be persons of independent mind, in my experience most intellectuals are conformists. If the smart people they know are indifferent to God, they are ashamed to take questions about God seriously. There arises a kind of groupthink. (4) I also remember that when I was an atheist, I didn't want God to be God; I wanted J. Budziszewski to be God. Insofar as intelligence is correlated in this fallen world with intellectual pride, and intellectual pride with atheism, we have another spurious correlation between intelligence and atheism.
Your first question is the next easiest. What difference does it make whether the physical habitation of humanity occupies a small or large part of the physical universe? Would it be more reasonable if we filled it up more completely? But why? C.S. Lewis used to say that we humans are too easily worried by sheer numbers, and too ready to abase ourselves before them. All those square light years of space disturb us because they seem so huge and empty. But why should we imagine them to be empty? Presumably God has uses for the rest of the physical universe; the fact that we don't know what these uses may be does not create a presumption against them, and in the meantime the best counsel is to admire its beauty. What is truly great is the Creator’s love; in what He has made, the greatest thing is not a galaxy or a cluster of galaxies, but a rational soul, made in his image, small though the body to which it is united may be.
As to the time the universe existed before God brought us on the stage -- that is nothing. To God, the passage of ten billion years is as the tick of a clock. Besides, we were made for eternity. We will be living, some eternally united with Him, some eternally alienated from Him, long after the suns of this universe have burned out or been replaced by the new creation.
Your second question is the most difficult, though it has an answer too. Allow me to begin with the observation that Christian philosophers and theologians have offered a variety of sophisticated responses to the problem of evil. I highly recommend Joseph Pieper, The Concept of Sin, which focuses on intentional evil, and Charles Journet, The Meaning of Evil, now out of print, which surveys all sorts of evil. The most important point, I think, is that God is under no obligation to make a world in which evil cannot happen; what He is impelled by His own goodness to do is make sure that evil will not have the ultimate victory, and that in the meantime, even those evils that He permits can be turned by His providence to good. Indeed it seems that a universe in which evil could never happen would be morally inferior to this one. How, for example, could one learn the great and angelic virtue of courage, unless there were things that we needed to be courageous about?
On the other hand, although these ‘blackboard’ solutions are very good, they may not be what the suffering soul is asking for. But there is another kind of answer too. Although God has not revealed to us all of the reasons why he permits our suffering, we know His attitude toward it, because He took the worst of it upon Himself for sheer love. It is this God who promises that in the New Jerusalem, He will personally will wipe the tears from the eyes of those who were afflicted, and for such a God, we can wait.