Why did God let sin into the world?
God has always given His creation free choice in whether or not to follow Him. His desire is that we choose to follow Him rather than just obeying pre-programmed instructions like robots.
The angels have that same free choice. Lucifer chose to try to lead his followers in rebellion against God to overthrow Him and take control himself. He failed in this and his insurrection was put down causing the destruction we see in Gen. 1:2 and Jer. 4. The creation itself fell from what God had originally created it to be. The rest of Gen. 1 records God restoring the Earth to a second habitable state and restoring life on the planet.
Mankind had dominion over the creation this time. Man was less powerful than the angels who had had dominion on Earth before. Mankind again had free choice. We don't know what restrictions man had other than the command to not eat of the tree of the knowledge of good and evil, but that one was recorded. Man had free choice, and made the wrong choice. It was not God's desire for man to make that choice, but He wanted a creation that consciously chose good so He gave them a test. They listened to Satan's lies (Gen. 3:4-5) that made God's restrictions sound bad - just like so many people do today about social topics that are really black and white sin according to God.
But He didn't forget us or stop loving us just because we failed Him. He gave Himself for us by having one of the three members of the Godhead temporarily set aside His divinity and be born of Mary to live and die as a sacrifice for our sins. He still wants you to make the right choice and choose that sacrifice to make yourself right with Him. It's the only acceptable sacrifice in His sight - and Christianity is the only religion in the world where the divine sacrificed Himself for His creation.
Some of what we call evil today is stirred up by Lucifer and his agents because he hates us and hates God's creation and hates God. Much of the rest of the evil is our own fault, because humanity now has that basic sin nature that is against God. We do what seems right to us, even though God has told us it is wrong. Eventually, Christ is coming back to take the Christians out of this world to be with Him. Then He is going to begin a period of judgment of the people behind that will last for around 7 years. They could still turn their lives over to Him by accepting Christ as Savior, but not many will. After that there will be a 1,000 year time period where He rules on Earth.
After all this time Lucifer and some of mankind will again rebel and try to win against Christ. This insurrection will be put down and the evil angels and the wicked dead will be judged and sent to the lake of fire for eternity - the second death (Rev. 20:7-15). The humans must still choose to do right. The Bible says that we will be able to look on those being punished eternally as an aid to make the right choice (Isa. 66:22-24). But we must still continue to make the right choice.