This is a personal letter from John to Gaius. All instances of "you" and "your" refer to Gaius and are singular.
This refers to John, the apostle and disciple of Jesus. He refers to himself as "elder" either because of his old age or because he is a leader in the church. The name of the author can be made explicit: "I, John the elder, am writing."
This is a fellow believer to whom John is writing this letter.
"whom I truly love"
"you may do well in all things and be healthy"
"just as you are doing well spiritually"
"fellow believers came." These people were probably all male.
Walking on a path is a metaphor for how a person lives his life. Alternate translation: "you are living your life according to God's truth"
John speaks of those he taught to believe in Jesus as though they were his children. This emphasizes his love and concern for them. It could also be that he himself led them to the Lord. Alternate translation: "my spiritual children"
John's purpose in writing this letter is to compliment Gaius in the way he took care of traveling Bible teachers; then he talks about two people, one evil and one good.
Here this is used as a term of endearment for a fellow believer.
"you are doing what is faithful to God" or "you are being loyal to God"
"help fellow believers and those you do not know"
John is thanking Gaius for his normal practice of helping these believers.
Here "the name" refers to Jesus. Alternate translation: "for they have gone out to tell people about Jesus"
receiving no gifts or help
Here "Gentiles" means non-Jews who do not believe in Christ.
Here the word "we" refers to John and those with him, and possibly includes all believers.
"so that we will cooperate with them in announcing God's truth to people"
The word "us" refers to John and those with him and does not include Gaius.
This refers to Gaius and the group of believers who met together to worship God.
He was a member of the congregation.
"who loves to be the most important one among them" or "who loves to act as though he's their leader"
"and how he says evil things about us that certainly are not true"
"did not welcome the fellow believers"
"stops those who want to welcome the believers"
"he forces them to leave the congregation"
Here this is used as a term of endearment for a fellow believer. See how you translated this in 3 John 1:5.
"do not copy the evil things that people do"
There are words left out but they are understood. Alternate translation: "but imitate what is good" or "but imitate the good things that people do"
"belongs to God"
"does not belong to God" or "does not believe in God"
Here "we" refers to John and those with him and does not include Gaius.
This can be stated in active form. Alternate translation: "All who know Demetrius testify that he is a good man" or "Every believer who knows Demetrius speaks well of him"
This is probably a man whom John wants Gaius and the congregation to welcome when he comes to visit.
"the truth itself speaks well of him." Here "truth" is described as a person speaking. Alternate translation: "everyone who knows the truth knows he is a good person"
What John is confirming is implied and can be made specific here. Alternate translation: "We also testify about Demetrius" or "We also speak well of Demetrius"
This is the end of John's letter to Gaius. He gives some final remarks and closes with a greeting.
John does not wish to write these other things at all. He is not saying that he would write them with something other than pen and ink.
John wants to visit him, but he does not know with certainty that he would be able to visit him. Alternate translation: "I plan to come to you" or "I want to come to you"
"Face to face" here is an idiom, meaning "in person." Alternate translation: "in person"
"May God give you peace"
"The friends here greet you"
"Greet each of the believers there for me"