First I will reccommend highly this site
Root - Answers in Genesis
They are much more educated than I am and have some fantastic ways of explaining things.
I was talking about this with a friend last night, about how we would respond to that very question. And I will try. I am by no means an expert, but I believe I have a simple grasp of some things about it. Certainly not all of it.
To truly understand why there is evil in the world and why God allows it we need to go back to Genesis. God created the universe. God created the world. He created man and woman in His image. Everything was good. Not just good, very good. Everything was perfect, there was no sin. There was only one thing Adam and Eve were suppose to do, not eat from the tree of the knowledge of good and evil. Essentially God was saying you have a choice. Obey me or disobey me (by eating from the tree). When they ate from the tree it was an act of obedience. But it was more than that. It was saying "We know better than you God, because the tree will allow us to know good and evil, we will become Gods ourselves."
Thus sin entered the world. Sin is the consequence of disobeying God. Because of His holy nature sin cannot be in the presence of God. Because of His just nature there is a punishment for sin, and that is death and eternal damnation in a place without God.
From that moment when sin entered, things began to break down and decay. God originally intended for things to be upheld and last eternally, but with sin we now have death, disease, decay. Humans now "grew old". A person starts dying from the moment they are born in a way.
God could have just said "you messed up" and just started over. But He didn't. He chose to clothe them when they were naked and ashamed. And even though they were punished, in Gensis 3:15 it is called the protoevangelion, or God is referring to the time when Christ will come and pay for our sins.
We all have inherited the sin nature from Adam and Even since we were born. You may think this is unfair, but think back to what happened and be honest with yourself. If you were Adam and Eve, would you really have done anything differently? Think of a choice you've made where it was clear what was right and what was wrong, but yet you went against God anyway. That is essentially the choice Adam and Eve made, none us of would have done any better.
Because God loves us, He made a plan (if you want to get deep into it, it was always His plan, but that's another discussion probably) to redeem us and make a way for the punishment of sin to be paid without us having to spend an eternity in hell. That way is through Jesus Christ.
People look at bad things happening and say "How could a loving God send an innocent person to hell?" But the truth is that is speaking from a human perspective. We are not innocent. We are sinners. We deserve eternal punishment because we've chosen to disobey God. It is only through God's mercy that He witholds judgement as long as He does, and there will come a day when judgement is no longer witheld. But for now, he chooses to withold it till the right time.
We think of love as being something related to the pain we are going through "right now". The pain of a loved one experiencing suffering relates to you right now. At this time. But God does not look at things in the same way we do. I have a list from one of my former pastors of 42 reasons God uses pain and suffering. While God did not choose to allow evil to come into this world, that was our fault as human beings, God can still choose to work through evil and for His glory. In God's eyes the loving thing to do is not let get rid of our pain, but help use it to draw closer to Him. Because that is His ultimate goal, that we should have a personal relationship with him. The long term benefit of having a relationship with Jesus Christ and your eternal salvation far outweighs the negatives of pain and suffering in God's eyes.
Say a man does kill a child. The family is devasted, unable to cope with the loss. A Christian friend who has lost a child comes to them and gives them hope because they too have lost a child and know the pain it brings but they found healing in Christ. (This is incredibly simplified but I'm just trying to make a point). They later come to accept Jesus as their savior. Was the child's death evil? Perhaps the child was saved before he died. It may have been evil in which the child was murdered, or the way in which the Christian friend's child died. But even through that evil, God can still work to good.
As far as children go, and especially abortions I don't have a clear answer. It talks in the bible about children having guardian angels, and God is watching over them. My personal answer would be that aborted babies go to heaven, unless God somehow gives them a chance to believe in Him (that would be hard to argue, I dont know). God has said though that everyone will have had a chance to believe in Him before going to hell.
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