I quite some time ago came to the conclusion that if God exists, he either does not interfere at all with the running of the universe, or is not omnipotent/omniscent, or is evil.
To take the OP's example of doomed children: Either God is capable of saving them or he is not. Religions are very firm on the subject of God's omnipotence (and indeed this is essential if he is to provide ultimate judgement in the future). If therefore he is capable of saving them, he either chooses to, or does not. To allow someone to die when you can save them is a despicable act, so if God chooses this course, he is evil in nature.
The only get out option I can see is if the universe is complete, and God no longer has any power to interfere, for good or ill (strongly against the teaching of most religions), and is merely omnipotent with respect to judgement and the afterlife. This would just about be morally acceptable to me, though I would require justification of why God is restricted in such a manner.
For similar reasons I am baffled by religious people who cite, and indeed look for the existence of miracles to prove the existence of God. While a genuine miracle would prove the existance of God, it would also prove that God is either of extremely poor moral fibre, or not omnipotent. If one person out of a thousand pilgrims is miraculouly cured of a crippled leg or something, why has this person been selected as superior to all the others? Either God is incapable of curing the others, or chooses not to, which I do not regard as morally acceptable. Odd though it may sound I would be very sad if it were proven that one of this type of miracle was genuine. In God's favour though, I have never seen such proof.
The test was a simple one, it could not be easier. Just do not from the fruit of one tree. I mean how hard was that? There would have been plenty of ther trees to eat from but it was only after Satan put doubt into their minds that they sinned. It was a simple test that they failed and now we are suffering from the results of their disobedience.
I have seen this line trotted out innumerable times by Christians, but if God is genuinely punishing for the actions of Adam and Eve (even assuming they exist) the world is an unbelieveably sick place. No one is responsible for actions committed by their ancestors, unless they knowingly took part in them. Consider an example; while Hitler's children did not survive, if they had, should we have stuck them on trial at Nuremberg and executed them? No one is responsible for any but their own actions.
In any case I regard most of the Old Testament as meaningless in reference to God. The "god" described in it is petty, childish, spiteful and hypocritical. It could learn morality from the average five year old. It is also very clearly neither omniscent nor omnipotent.
You're assuming that the suffering people you witness are actually people, and not automatons placed there by God to give you a moral conundrum.
Really, you THINK there is a suffering person, and you're morally obligated to act. You can choose to, or choose not to. Your choice. And you'll be judged on your choice.
But you think God would ACTUALLY allow unnecessary suffering? Not when he could just put a Turing-capable automaton there to make you THINK there is suffering.
The suggestion that the world exists purely for one individual's benefit and is otherwise populated by soulless automatons is an interesting one, El Machinae. However, surely God would then require us to know that fact? Otherwise he merely creates the impression that it is acceptable for him to murder, torture, and neglect huge suffering, and so would erroneously impy it was acceptable for us.
To allow the torturing of infants is wrong
Even for a Greater Good?
For humans with their limited power, possibly, and within reason, such as the very mild examples you suggest. For an omnipotent being, no. By definition he can find a solution with all the benefits and none of the downsides, and knows it as he is omniscent.