A perfect man, Jesus, took the place of the perfect man, Adam. "For as in Adam, all die, so in Christ, shall all be made alive." 1 Corinthians 15:22. The principle that you're asking about is right in that verse. We all die, because we inherited sin. We cannot overcome sin because we were "…brought forth in iniquity, and in sin did my mother conceive me [us]" Psalm 51:5. Thus we need Jesus to redeem us: 1 Timothy 2:6, "who gave Himself a ransom for all to be testified in due time.”
In the present, do Christians sin after they are converted? Yes. (See 1 John 1:10.) How do we receive forgiveness then? 1 John 1:9, “If we confess our sins, he is faithful and just and will forgive us (Christians) our sins and purify us from all unrighteousness.” Through prayer, God wants us to confess our sins and ask for forgiveness through Jesus’ blood. However, that’s not the end of the process. God will continue to work with us to cleanse us from sin’s defilements. God is so kind. He is constantly working to develop our characters into Christlikeness.
What about future sins? If faithful, when Christians are resurrected we will have perfect, spirit bodies. We will not sin; we will be pure. 1 John 3:2, 3, “…we know that when he appears, we shall be like him…everyone who has this hope in him purifies himself, just as he is pure.”
Does does Jesus’ sacrifice apply to the world? Yes. Christ’s atoning blood is what guarantees the resurrection, education and development of the whole world. Recall 1 Timothy 2:4-6, God “desires all men to be saved and to come to the knowledge of the truth. For there is one God and one Mediator between God and men, the Man Christ Jesus, who gave Himself a ransom for all, to be testified in due time”. During Christ’s 1,000 reign, the testimony that Jesus’s ransom sacrifice covers all will be given. Then “…the inhabitants of the world will learn righteousness.” (Isaiah 26:9) At the end of the 1,000 years, Satan will try to deceive the nations again. Those who fail this test will go into second death. But the great majority of mankind will be perfect human beings who will sin no more. Revelation 21:27, “But there shall by no means enter it anything that defiles, or causes an abomination or a lie, but only those who are written in the Lamb’s Book of Life.”