Answer: Actually, there is more than one question in that one question! Thankfully, the Bible answers all of them.
On the broadest of scales, yes, one could say that God is responsible in a way for what happens in this world. As a wise man once said, the most important fact in the universe is that “God reigns supreme!” The first book of the Bible tells us God made the world, and the last book of the Bible tells is that He will wrap all things up after achieving His plan: A glorified, divine family filled with His children who forever experience life just as He does. The Bible says that when that time comes, all of us who are there will look back on all we experienced in the world, as terrible as it seemed at the time, and conclude that it was nothing but a small price to pay for the eternal, glorified existence we will then know. (Don’t take our word for it: Check out Romans 8:18-25.)
Yet, at a smaller and more intimate—more real—scale: No, God is not responsible for the individual things that happen in the world. The evils of this world are due to the sins of this world, committed by individuals with free will (frankly, including you and me). Mankind had the opportunity to live in a world ordered and upheld by God and our original founding fathers (actually, father and mother, Adam and Eve) threw it all away to run the world themselves, their own way. (It’s depressing, but you can read it for yourself in Genesis 2 and 3.) And, honestly, we’ve all made similar choices in our lives, just like they did: Choosing our way instead of God’s. When we choose to run the world our own way, can we blame God for how it turns out?
Though, soon, God is going to send Jesus Christ to remove the devil who tempted our forefathers to sin and who tempts us today (Much less depressing! Read that in Revelation 20:1-3.) Then He will lay the foundation for a better world, in which “the earth shall be full of the knowledge of the Lord as the waters cover the sea” (Isaiah 11:9).
So, putting the answers together: Since the Garden of Eden, thousands of years ago, we are responsible for what happens in this world, as is the devil, whom the Bible calls the “god of this age” (2 Corinthians 4:4) and the “ruler of this world” (John 12:31; 14:30; 16:11). But eventually, mankind will learn the lessons of its entire history. (Think we’ve learned them, yet? Watch the news. Enough said.) That lesson will be that we need God—His guidance, love, and mercy. And He will take all we’ve been responsible for, and He will make right all that we have done wrong.