Jesus’ words in Matthew 5:27-30 single out adulterers, persons who lust after what is not theirs. When he says “If your right eye causes you to sin” he means looking lustfully after a woman. When he says “if your right hand causes you to sin” Jesus is still talking about committing adultery, which is a sin.
Outside of Jerusalem was the valley of Hinnom where the bodies of murderers and criminals were thrown to be consumed by the flames.
The word gehenna is the Greek transliteration of the Hebrew ge-hinnom, meaning “Valley of [the sons of] Hinnom.” In later years, Gehenna continued to be an unclean place used for burning trash from the city of Jerusalem. Jesus used Gehenna as an illustration of hell. The Gehenna Valley was a place of burning sewage, burning flesh, and garbage. Maggots and worms crawled through the waste, and the smoke smelled strong and sickening (Isaiah 30:33). It was a place utterly filthy, disgusting, and repulsive to the nose and eyes. Gehenna presented such a vivid image that Christ used it as a symbolic depiction of hell: a place of eternal torment and constant uncleanness, where the fires never ceased burning and the worms never stopped crawling (Matthew 10:28; Mark 9:47–48). https://www.gotquestions.org/Gehenna.html
In Matthew 10:28 Jesus went on to say that we should not fear those who are capable of killing the body, but to be afraid of Him who can destroy both body and soul in hell (Gehennah). As for the human soul, Ecclesiastes 12:7 says that our dead bodies return to dust, there to remain in the earth, but the spirit returns to God. https://www.gotquestions.org/Bible-verses-about-the-soul.html
After Jesus died, it was His body that was buried in the tomb. However, Jesus’ spirit/soul was not in the tomb. Jesus’ spirit was in the Father’s presence (Luke 23:46; Ephesians 4:8).
When Jesus was dying on the cross and one of the thieves being crucified with Him asked Him for mercy, Jesus replied, “I tell you the truth, today you will be with me in paradise” (Luke 23:43). Jesus knew that His death was imminent and that He would soon be in heaven with His Father. Therefore, Jesus used “paradise” as a synonym for “heaven.” The apostle Paul wrote of someone (probably himself) who “was caught up to paradise” (2 Corinthians 12:3). In this context, paradise obviously refers to heaven.
Both paradise [heaven] and hell are “temporary holding places” until the day when Jesus Christ comes back to judge the world based on whether or not individuals have believed in Him. The first resurrection is of believers who will stand before the Judgment Seat of Christ to receive rewards based on meritorious service to Him. The second resurrection will be that of unbelievers who will stand before the Great White Throne Judgment of God. At that point, all will be sent to their eternal destination—the wicked to the lake of fire (Revelation 20:11-15), and the righteous to a new heaven and a new earth (Revelation 21—22). https://www.gotquestions.org/paradise.html
You ask if it is the body or the soul that enters into heaven or hell. According to mainstream Christianity, the body goes into the ground and returns to dust. The human soul awaits the resurrection when those whose names are written in the book of life are clothed with glorified bodies, bodies fit to spend eternity in heaven. Our example is Christ Jesus, who was resurrected with a glorified body and ascended into heaven and even now is seated at the right hand of God.