Who wrote the book of John?

In my study, I started comparing the 4 gospels of Matthew, Mark, Luke and John because I wanted to see how each captured the instructions Jesus gave the disciples when he sent them out giving the the authority over demons.  I found it interesting that He sent them out with no money, no bags, only one coat and sandals:

 ⁸These were his instructions: "Take nothing for the journey except a staff—no bread, no bag, no money in your belts. ⁹Wear sandals but not an extra tunic. ¹⁰Whenever you enter a house, stay there until you leave that town. ¹¹And if any place will not welcome you or listen to you, shake the dust off your feet when you leave, as a testimony against them."

When I got to John, I started looking at when Jesus raised Lazarus from the dead.  Amazing story, right?  So, why is it NOT in the other gospels?  So in John 11-

 Now a certain man was sick, Lazarus of Bethany, the town of Mary and her sister Martha. ² It was that Mary who anointed the Lord with fragrant oil and wiped His feet with her hair, whose brother Lazarus was sick. ³ Therefore the sisters sent to Him, saying, “Lord, behold, he whom You love is sick.”

"He Whom You love".. interesting use of words I thought, so I looked further.. the only other times I can find reference to "The one whom He loved" was at the last supper- John 13 -

²³ Now there was leaning on Jesus’ bosom one of His disciples, whom Jesus loved.

²⁴ Simon Peter therefore motioned to him to ask who it was of whom He spoke. 
²⁵ Then, leaning back on Jesus’ breast, he said to Him, “Lord, who is it?” 

And on the cross - John 19-

²⁵ Now there stood by the cross of Jesus His mother, and His mother’s sister, Mary the wife of Clopas, and Mary Magdalene. ²⁶ When Jesus therefore saw His mother, and the disciple whom He loved standing by, He said to His mother, “Woman, behold your son!” ²⁷ Then He said to the disciple, “Behold your mother!” And from that hour that disciple took her to his own home.

So, who was this disciple "that Jesus Loved"?  I have always been told it was John. But would it make sense that it was actually Lazarus?  I am not claiming some great revelation here, just a thought.  

Couple of thoughts, John was not considered very educated, could he write such an eloquent book?  Sure he could with the holy Spirit guiding him, but I do not remember any other act of the Holy Spirit making someone literate.  Other thought, the mention of the disciple at the cross "the disciple whom He loved", I believe from what I have read, this was the only follower or disciple at the actual cross while Jesus was being crucified.  Who better than Lazarus to be there and not be afraid of being linked to Jesus and be killed along with Him?  He had already been dead once and knew what it was like and knew that Jesus was in fact God and could protect him or even bring him back to life again if need.  The book of John mostly covers Jesus's ministries in and around Judea.. that is were Lazarus lived (and died and lived again).  

Does it matter?  Not really I guess, but I do find it interesting to ponder.  I mean, think about it, if this book was actually written by Lazarus, it means it could be a much better insight to life after death since Lazarus actually, for a better term, lived it. 

I know a lot of times when I think of the "disciples" I think of just the twelve we know of, but certainly, the term "disciple" at least in the book of John, was anyone who followed Jesus as eluded to in John chapter 6 - 

 ⁵³Jesus said to them, "I tell you the truth, unless you eat the flesh of the Son of Man and drink his blood, you have no life in you. ⁵⁴Whoever eats my flesh and drinks my blood has eternal life, and I will raise him up at the last day. ⁵⁵For my flesh is real food and my blood is real drink. ⁵⁶Whoever eats my flesh and drinks my blood remains in me, and I in him. ⁵⁷Just as the living Father sent me and I live because of the Father, so the one who feeds on me will live because of me. ⁵⁸This is the bread that came down from heaven. Your forefathers ate manna and died, but he who feeds on this bread will live forever." ⁵⁹He said this while teaching in the synagogue in Capernaum. ⁶⁰On hearing it, many of his disciples said, "This is a hard teaching. Who can accept it?" 

So, I welcome your thoughts.... and God Bless you for listening to mine as I attempt to be a better disciple. 

Carl