What does it Mean to Act in the Name of Jesus?

What is the power of someone's name? It is a word or expression in which someone is represented to us. When I mention or hear a name, it brings to mind the whole man, what I know of him, and also the impression he has made on me. The name of a king includes his honor, his power, and his kingdom. His name is the symbol of his power. And so each name of God embodies and represents some part of the glory of the Unseen One. The Name of Christ is the expression of everything He has done and everything He is and lives to do as our Mediator.
What does it mean to do a thing in the name of another? It is to come with his power and authority, as his representative and substitute. Using another's name always presupposes a common interest. No one would give another the free use of his name without first being assured his honor and interests were as safe with another as with himself.
What does it mean when Jesus gives us power over His Name--the free use of it--with assurance that whatever we ask in His Name will be given to us? (John 14:13-14; 15:16; 16:23-24, 26). Jesus solemnly gives to all His a general and unlimited power to use His Name at all times for everything they desire. He could not do this if He did not know that He could trust us with His interests and that His honor would be safe in our hands.
The free use of someone else's name is always a token of great confidence and close union. Someone who gives his name to another stands aside to let that person act for him. Someone who takes the name of another gives up his own as of no value. When I go in the name of another, I deny myself. I take not only the name of another, but himself and what he is, instead of myself and what I am.
When the Lord Jesus went to heaven, He left His work--the management of His Kingdom on earth--in the hands of His servants. He also gave them His Name to draw all the supplies they needed for the due conduct of His affairs. Christ's servants have the spiritual power to use the Name of Jesus only insofar as they yield themselves to live only for the interests and the work of the Master. The use of the Name always supposes the surrender of our interests to Him Whom we represent.
Oneness of life on earth gives oneness of name. We are one with Jesus; we have one life and one Spirit with Him. For this reason we may proceed in His Name. Our power in using that Name, whether with God, men, or devils, depends on the measure of our spiritual life-union with Christ. Our use of His Name rests on the unity of our lives with Him.
No one really gives himself up to live in the Name of Jesus without receiving in ever-increasing measure the spiritual capacity to ask for and receive in that Name whatever he desires. My bearing of the name of another shows that I have given up my own name and, with it, my own independent life. But just as surely, it shows I have possession of everything belonging to the name I have taken instead of my own.
We are not acting in the name of someone who is absent. Jesus Himself is with the Father. When we pray to the Father, it must be in Jesus' Name. The Name represents the person. The Name and the Spirit of Jesus are one. To ask in His Name is to ask in full union of interest, life, and love with Himself, as one who lives in and for Him.
John 14:13, "And whatsoever ye shall ask in my name, that will I do," means "in my nature." With God, things are requested according to their nature. Asking in Christ's Name doesn't mean that at the end of some request we say, "This I ask in the Name of Jesus Christ." It means we are praying according to His nature, which is love that doesn't seek its own will, but only the will of God and the good of all creatures. Such asking is the cry of Christ's own Spirit in our hearts.

As we bear the Name before men, we have the power to use it before God. Let us plead for God's Holy Spirit to show us what the Name means, and what the right use of it is. It is through the Spirit that the Name, which is above every name in heaven, will take the place of supremacy in our hearts and lives. The treasures and powers of the spiritual world are placed at your disposal to help those around you.

Must we address Jesus by his Name?
The name "Jesus" is the name associated with "the shame" which he endured in order to "save his people from their sins" (Matthew 1:21). Therefore, his people (those who had a personal relationship with him) never addressed him as "Jesus," but as "master" or "lord" (John 13:13-14, Luke 6:46), and so should all his people today.
The only ones who irreverently called him "Jesus" were demons, his enemies, and those who did not have a personal relationship with him. We should not follow the examples of them, but we should follow the examples of those who had a close, personal relationship with him. Those who had a personal relationship with Jesus never addressed Jesus, face to face, as "Jesus."
The following passages are all the verses in scripture in which Jesus was personally addressed as "Jesus," face to face. Notice who was addressing him, face to face, in each case.

A man possessed with devils (Matthew 8:29)
A man with an unclean spirit in the synagogue (Mark 1:23, Luke 4:34)
A man with an unclean spirit in the country of the Gadarenes, (Mark 5:7, Luke 8:28)
Jews in a crowd (Matthew 21:11)
Unbelieving Jews (John 6:42)
The accusers of Peter, who were enemies of Christ (Matthew 26:69,71, Mark 14:67)
Pilate, the one who delivered Jesus to be crucified (Matthew 27:17,22,37, John 18:33, John 19:19)
Chief priests and elders, who persuaded the multitude to destroy Jesus (Matthew 27:20).

Did anyone else call him Jesus? Yes. Some of the people Jesus healed called him "Jesus," but they had no personal relationship with him at that time.

Lepers called him "Jesus" (Luke 17:13).
A man who was sick for 38 years told others that "Jesus" healed him (John 5:15).
Bartimaeus, who was blind, called him "Jesus" (Mark 10:47, Luke 18:37-38). But notice the reason why he called him "Jesus." It was because when he asked what the commotion in the crowd was all about, a Jew told Bartimaeus that "Jesus" was passing by.

Notice the following three parallel passages. Jesus asked his disciples who they thought he was. Peter did not say, "Thou art Jesus." Instead, Peter called him according to the relationship between the two of them.

Matthew 16:13-16, "When Jesus came into the coasts of Caesarea Philippi, he asked his disciples, saying, Whom do men say that I the Son of man am? And they said, Some say that thou art John the Baptist: some, Elias; and others, Jeremias, or one of the prophets. He saith unto them, But whom say ye that I am? And Simon Peter answered and said, Thou art the Christ, the Son of the living God."
Mark 8:27-29, "And Jesus went out, and his disciples, into the towns of Caesarea Philippi: and by the way he asked his disciples, saying unto them, Whom do men say that I am? And they answered, John the Baptist: but some say, Elias; and others, One of the prophets. And he saith unto them, But whom say ye that I am? And Peter answereth and saith unto him, Thou art the Christ."
Luke 9:18-20, "And it came to pass, as he was alone praying, his disciples were with him: and he asked them, saying, Whom say the people that I am? They answering said, John the Baptist; but some say, Elias; and others say, that one of the old prophets is risen again. He said unto them, But whom say ye that I am? Peter answering said, The Christ of God."
Therefore, when someone asks you, "Who are you?" do not tell them your name, but tell them who you are according to the Word and Spirit of God. You are a bondman of Christ.

Your Questions Answered

But Christ told the apostle Paul that his name was "Jesus." Therefore, it must be okay to call him "Jesus."

Acts 9:5, "And he said, Who art thou, Lord? And the Lord said, I am Jesus whom thou persecutest:"
Acts 22:8, "And I answered, Who art thou, Lord? And he said unto me, I am Jesus of Nazareth, whom thou persecutest."
Answer: But the apostle Paul was not an apostle at this time. His name was not even "Paul," it was "Saul." He was an enemy of Christ at the time. As you can see from this same passage, Saul was "persecuting" Christ by persecuting believers in Christ.

What about the parents of Jesus? Did they not call him Jesus?
Answer: No, they did not call "him" Jesus, they called "his name" Jesus.
Concerning Joseph:

Matthew 1:21, "And she shall bring forth a son, and thou shalt call his name JESUS: for he shall save his people from their sins."
Matthew 1:25, "And knew her not till she had brought forth her firstborn son: and he called his name JESUS."
Concerning Mary:

Luke 1:31, "And, behold, thou shalt conceive in thy womb, and bring forth a son, and shalt call his name JESUS."
Concerning both Joseph and Mary:

Luke 2:21, "And when eight days were accomplished for the circumcising of the child, his name was called JESUS, which was so named of the angel before he was conceived in the womb."
Notice in the above passages, it does not say they called "him" Jesus, it says they called "his name" Jesus. There is a difference. There is not one scriptural passage where they began talking to their son with, "Jesus..."

But what about the many times in scripture where the epistles refer to him as "Jesus?"
Answer: It is okay to refer to Christ as "Jesus" in the 3rd Person (when speaking about Christ), but not in the 2nd person (when speaking to Christ directly).
For example, when Jesus appeared to his disciples during his resurrection, his disciples did not know it was Jesus, and they talked to him about "Jesus" (Luke 24:19). But once they realized it was Jesus they were talking to, they addressed him as "lord" and "master."
When believers were talking to one another about Jesus, they would address Christ as "Jesus," but these same people would never call Christ "Jesus" while addressing him to his face, while in his presence. For example, Philip called Christ "Jesus" when speaking to Nathaniel (John 1:45) about Jesus, but Philip never called Christ "Jesus" when speaking to Christ.
The same with angels. When angels talked to believers, they would refer to Christ as "Jesus" (Matthew 28:5, Mark 16:6, Acts 1:11), because they were speaking about Jesus to another. And the same goes for all the epistles in scripture. The epistles speak of "Jesus" when talking about Jesus, but never does anyone with a personal relationship with Christ call him "Jesus" when speaking to Christ.

Conclusion
When we "ask in His name" and talk to the Christ today, we must remember that "Jesus" is more than "a name." He is our Lord and Master, and we should address him as such. Because the name, whether it be Jesus, Joshua, or Yeshua, equally means, "YHVH is our salvation."