According to Catholicism, can the dead in Christ affect this world?

The story of the Rich Man and Lazarus suggests, "No." Travel and communication between the realms of life and death is neither permitted (laws against using mediums and spiritists) nor possible (absent God's intervention, such as with Enoch and Elijah).

"And besides all this, between us and you a great chasm has been set in place, so that those who want to go from here to you cannot, nor can anyone cross over from there to us." (Luke 16:26)

The story of the souls under the altar suggests, "Yes!"

9 When he opened the fifth seal, I saw under the altar the souls of those who had been slain because of the word of God and the testimony they had maintained. 10 They called out in a loud voice, “How long, Sovereign Lord, holy and true, until you judge the inhabitants of the earth and avenge our blood?” 11 Then each of them was given a white robe, and they were told to wait a little longer, until the full number of their fellow servants, their brothers and sisters, were killed just as they had been. (Revelation 6:9-11)

Those souls under the altar have not yet been resurrected, as the resurrection of the dead in Christ and glorification of those still alive at the time of Christ's coming happens at the same time, according to Saint Paul, as written in 1 Thessalonians 4:15. Yet those souls are able to speak to the Sovereign Lord in prayer and petition to Him to execute justice. Furthermore, their petition, delivered via John before they even lived, has come to our ears. Their words, found in Holy Scripture can reach us, teach us, and move us. Thus through the intermediating office of the Apostle John and the Word, such souls are part of our lives.

One more piece of evidence suggests another yes. It concerns the authority delegated to the Virgin Mary. The Gospel of John and the Book of Revelation are curious in their construction. They form a chiasm with one another. If you compare the seven sign miracles of John to the division of Revelation into pieces at the places where thunder is heard (other than the speech of the Seven Thunders), there is a marvelous pattern. (The fourth and fifth miracles swap places, but there is warrant for this from Job 3 and other places where the 4th and 5th parts of a group of seven are intentionally swapped. Too much detail to get into here.)

The chiasm has these matches. The Revelation reference is to where the thunder verse occurs. The events from Revelation are from its corresponding section, which may span several chapters.

• 1st miracle: Wedding at Cana, changing water into wine.
• 7th thunder: Rev 19:6 Wedding supper of the Lamb.

What was Mar's role?

• 2nd miracle: Heal official's son; near story of the Samaritan woman at the well.
• 6th thunder: Rev 16:18. The harvest of the earth.

This harvest includes people from all cultures, including Samaritans.

• 3nd miracle: Paralytic is healed and told to stop sinning.
• 5th thunder: Rev 14:2. Behemoth & Leviathan, agents of religious and political oppression.

The man who was healed complained that he could not get to the pool of water in time when the angel of healing appeared. Leviathan frolics in water and Behemoth bathes there (in Job's account). Then when he picked up his mat, the religious authorities get upset.

• 4th miracle: Feeding the 5,000.
• 3rd thunder: Rev 8:5. The Great Multitude. The famine of the 3rd Seal.

The Great Multitude of martyrs compares to the crowd that is fed, but the unsaved world gets a famine instead of food.

• 5th miracle: Walking on water.
• 4th thunder: Rev 11:19. Mountain cast into the sea, poisonous wormwood cast upon the waters.

Jesus walks on water and calms the anxiety of his disciples, telling them to not be afraid. But the plagues on the world cause fear.

• 6th miracle: Heal a man born blind.
• 2nd thunder: Rev 6:1. Living creature with many eyes.

The blind man is accused of being a sinner and deserving his blindness, just like Job was. John is told not to weep because Jesus was found worthy, another use of ones eyes.

• 7th miracle: Raise Lazarus from the dead.
• 1st thunder: Rev 4:5. Jesus is called the firstborn from the dead.

This also mentions the book of life.

There are more points of comparison between the miracles and Revelation, but that would make this answer too long.

With these comparisons, we see that Mary, mother of Jesus, directs him to take care of the wine situation at the Wedding at Cana. Jesus says that it is not his time yet. Elsewhere, Jesus says he does not know when he will return, because the Father determines the time when he will return. Here we see his mother determining the timing of the start of his ministry. By parallel comparison to the Wedding Supper of the Lamb, it is logical to assume that Mary will play a part in setting the time for the ending of Christ's ministry. After all, ALL mothers care a lot about their children's weddings! Scripture says that Mary's heart would be pierced with a sword (the Word) in a distinct way. Swords are significant in Scripture and relate to the Word of God. This implies that special knowledge was given to Mary. If Jesus was subject to Joseph and Mary when he followed them from the temple as a child, it seems that the timing of Jesus' second coming - which is necessitated by showing mercy toward the suffering saints - will reflect Mary's compassionate will as well as that of God the Father.

The OP asks about both Catholics and Catholicism:

Catholics

A Pew Research Center survey implies that the answer would be yes, at least as far as US Catholics are concerned. First the general findings:

About seven-in-ten Americans say it is possible to feel “the presence of someone who has died,” while roughly half say that living people can be helped by those who have passed (51%) or communicate with them in some way (47%). When asked about their personal experiences with the deceased, 44% of U.S. adults say that they have felt the presence of someone who has died, while smaller shares say that they have received help from (20%) or communicated with (14%) someone who has died.

More specifically, Catholics are more likely to believe it is possible to get help from the departed (68% vs. 49% of Protestants) and communicate with them (57% vs. 46%).

Catholicism

One reason for the larger percentages among Catholics is surely the doctrine of the Intercession of Saints, which encourages believers to pray for and expect the help of those in heaven. Among the scriptural bases for praying to saints, the Catholic Answers web site says:

• The reason we pray to the saints is that they are still members of the Body of Christ. Remember, the life which Christ gives is eternal life; therefore, every Christian who has died in Christ is forever a member of the Body of Christ. This is the doctrine which we call the Communion of the Saints. Everyone in Christ, whether living or dead, belongs to the Body of Christ.

• The Bible encourages Christians to approach the saints in heaven, just as they approach God the Father and Jesus Christ the Lord: “But you have approached Mount Zion, the city of the living God, the heavenly Jerusalem, and myriads of angels, and the assembly and church of the firstborn who have been enrolled in heaven, and God the judge of all, and spirits of righteous ones who have been made perfect, and Jesus, the mediator of a new covenant, and the sprinkled blood which speaks better than that of Abel” (Heb. 12:22-24).

• in Revelation 8:3-4 we are told that something similar happened when the Lamb opened the seventh seal of the book: “Another angel came and stood on the altar, having a golden censer, and many incenses were given to him, in order that he will give it with the prayers of all the saints on the golden altar before the throne. And the smoke of the incenses went up with the prayers of the saints from the hand of the angel before God.”

We should also keep in mind that in Catholicism, angels are included among God's saints, sharing with human saints in the Beatific Vision. Thus any scripture that involves the appearance of an angel can serve as a basis for the belief that saints in heaven interact with and help believers.

According to Catholicism, can the dead in Christ affect this world?

The short answer is yes.

Throughout the history of Church those who have died in Christ (saints) have in one way or another had some affect on the world.

Even Sacred Scriptures have pointed this out occasionally too.

In 2 Kings 13, we can How the bones of the Prophet Elisius raised a person from the dead. It can be presumed this not only affected the one raised from the dead, but the whole community at large. This example is not strictly one as those died in Christ, but in God’s Grace prior to Christ’s coming into the world.

And Eliseus died, and they buried him. And the rovers from Moab came into the land the same year.

And some that were burying a man, saw the rovers, and cast the body into the sepulchre of Eliseus. And when it had touched the bones of Eliseus, the man came to life, and stood upon his feet. - 2 Kings 13:20-22

The Transfiguration of Christ seems to be yet another example, where Elijah and Moses interact with Jesus in the flesh, as well as with Peter, James, and John:

2 And after six days Jesus taketh with him Peter, and James, and John, and bringeth them up into a high mountain apart by themselves: and he was transfigured before them;

3 and his garments became glistering, exceeding white, so as no fuller on earth can whiten them.

4 And there appeared unto them Elijah with Moses: and they were talking with Jesus.

5 And Peter answereth and saith to Jesus, Rabbi, it is good for us to be here: and let us make three tabernacles; one for thee, and one for Moses, and one for Elijah.

6 For he knew not what to answer; for they became sore afraid.

7 And there came a cloud overshadowing them: and there came a voice out of the cloud, This is my beloved Son: hear ye him.

8 And suddenly looking round about, they saw no one any more, save Jesus only with themselves. - Mark 9:2-8

One saint in particular, brings out the fact that the saints can affect the world we live in. That saint is St. Joan of Arc, who constantly heard the voices of St. Michael the Archangel, St. Margaret of Antioch, and St. Catherine of Alexandria. These voices which she never doubted informed her of a special mission given her by God to crown the rightful king of France, King Charles VII.

At the age of 13, Joan of Arc had locutions — an interior, mystical phenomenon that involves hearing a divine voice — and reportedly heard the voices of St. Michael the Archangel, St. Margaret of Antioch, and St. Catherine of Alexandria. These three informed her of a special mission given her by God to crown the rightful king of France and thereby end the dynastic dispute that undergirded the Hundred Years’ War.

Along the way, she convinced lords, soldiers, and the French heir to the throne, Charles VII, of her mission. After a lengthy interrogation, she was given charge of the army and successfully lifted the siege of Orléans — on which the fate of the entire war hung — and then freed several towns along the route to crowning Charles VII in the cathedral of Rheims.

The story of Joan of Arc is true and historically documented. For this reason, she is among the most famous heroines of history. The task given her by God was so exceptional that it would lead atheist Mark Twain, who wrote a book on her life, to earnestly but exaggeratedly call her “by far the most extraordinary person the human race has ever produced.” - Who was Joan of Arc?: Answers to your questions about this heroic saint

Seeing to the fact that St. Joan of Arc was able to get the legitimate heir to the French crown back on the throne and chased the English out of France though the help of those who had died in Christ (saints ) as well as St. Michael the Archangel was extremely remarkable.

Although Joan of Arc saw these saints in visions, they were as real to her as those people we speak to in our everyday life!

Although Joan of Arc’s body was incinerated at the stake by the English, her heart remained intact after her execution. It could not be burned even after several attempts. The soldiers threw the heart in the Seine River so that no one would be able to venerate her remains. Would that very heart that so loved France be once again found to restore the French monarchy?