Yes-----But. Everything is a Yes but or a No but if you think about it. In this case the answer is Yes, God can be good and exist in the world---but. But we have to define who God is and but we have to define our relationship to who this God is. 

You see for most, when we see the pain and suffering of the world and we claim that God can't be good if the world looks like it does--usually what is the image of God in the brain at that moment? What is the view of God? Usually it is a pretty generic fuzzy big-man-in-the-sky mentality. There is no personality connected to this God. There is no history of what he has done in the past. There is no character that we can know him by. Therefore, its pretty natural to question his goodness.

Christians though historically have claimed God was good. Perhaps you have heard of the true story of Allen Gardiner.

 In 1852 there was a pretty important English missionary named Allen Gardiner. He was lost at sea for a while, so a search party went out to find look for him. The search party found him and his companions shipwrecked on Tierra del Fuego. Their provisions had run out and they had all starved to death alone, far from their families.  However, they found Allen's body and his journal. Inside Gardiner wrote about the pangs of thirst that were intolerable. Still throughout the journal, amazingly, he wrote out Scripture. He listed Psalm 34:10—“The young lions do lack, and suffer hunger: but they that seek the Lord shall not want any good thing.” More amazing still was what he wrote at the end of his journal, when his handwriting was feeblest and faint. Gardiner managed to write one last entry: “I am overwhelmed with a sense of the goodness of God.”

 Now how did a man, who was shipwrecked, starving, hungry, and in the pangs of thirst that are unimaginable---when he was feeling the worst—say he thought God was good? And how is it that so many people say God is not Good when they have suffered far less? What is the difference? 

The answer, I think, is comes down to how we think God's character really is. Christians have always said God was good, and as their source they used Scripture which not only states God's goodness but also shows it in the deeds God does. The main trajectory of the entire Scriptures is a God who comes after, dies, and saves a wayward people who does not love Him, and does not care for him. This starts with God caring for Adam and Eve in the garden, but comes to a fruition in Christ's work on the cross. Christians then point to the cross and say, "I might not know why I am shipwrecked and starving to death, but it can't be because God isn't good and doesn't love me because look what he has done for me on the cross, dying for me, saving me from myself." In other words, because Christians claim that they know the character of God based in his deeds, they don't conclude that he is not good.

On the other hand, if God is our minds has no history, and has no character we see him merely as a Santa Claus God who is suppose to reward us when we are good, and punish us when we are bad. This is overly simplistic and doesn't treat God any better than a puppeteer. So the question we have to ask before, "Is God good?" must be, "What is God like?" That latter question will end up answering the former.  

 There is a great illustration from the book—“The Life of Pi” where the main character is exploring Christianity and trying to figure it out. He talks to a priest named Father Martin who says the most important thing in the Christian faith is God coming into the world to die on the Cross.

 Pi thought: That a god should put up with adversity, I could understand. The gods of Hinduism face their fair share of thieves, bullies, kidnappers and usurpers. What is the Ramayana but the account of one long, bad day for Rama? Adversity, yes. Reversals of fortune, yes. Treachery, yes. But humiliation? Death?  I couldn’t image Lord Krishna consenting to be stripped naked, whipped, mocked, dragged through the streets and, to top it off, crucified—and at the hands of mere humans, to boot. I’d never heard of a Hindu god dying. Devils and monsters did, as did mortals, by the thousands and millions—that’s what they were there for…But the divinity should not be blighted by death. It’s wrong….It was wrong for this Christian God to let his avatar die. That is tantamount to letting a part of himself die. For if the Son is to die, it cannot be fake. If God on the Cross is God shamming a human tragedy, it turns the Passion of Christ into the Farce of Christ. The death of the Son must be real. Father Martin assured me that it was. But once a dead God, always a dead God, even resurrected. The Son must have the taste of death forever in His mouth. The Trinity must be tainted by it; there must be a certain stench at the right hand of God the Father. The horror must be real. Why would God wish that upon Himself? Why not leave death to mortals? Why make dirty what is beautiful, spoil what is perfect? Love. That was Father Martin’s answer.

What is the character of your God? If you don't believe in God, then how can you talk about "goodness" at all if we are just the sum total of atoms and electrical charges that have been developed over millions of years with no greater purpose then to pass our genes on to someone else? In a world like that goodness is false, and reproduction is ultimate. So what are you going to believe as everyone is believing (re: hoping and trusting in something)?