The confusion comes from Jesus switching meanings on Peter. This is what is called the last supper and they are all arriving in the Upper Room from the dusty streets below. Since they didn't normally use chairs in those days eating dinner with 13 guys with unwashed feet isn't a very appetizing prospect. When they arrive the Apostles are arguing amongst themselves about who is the greatest, they had expressed confusion about this sort of thing before. Jesus realizing that they had a problem with vain ambition teaches them a lesson in humility, by washing their feet, which is the role of a servant.
When Jesus got to Peter, Peter simply refused to let Jesus wash his feet. Peter had an argument with Jesus over this very thing once before and Jesus called him Satan, that is how serious Peter's problem was. You have to understand that the greatest in the Kingdom of heaven is servant to all, as opposed to a worldly concept where everyone serves the king. So Jesus whom Peter fully realizes is King takes on the role of a lowly role of a servant and Peter is rebuking Jesus because the King is doing something Peter feels is beneath his station in life. Jesus tells Peter that I am either going to wash your feet or you can get out!
Peter panics and says, and mind you I'm paraphrasing, No Lord wash all of me. What Jesus says is kind of humorous if you really think of it, He says, but Peter only your feet are dirty why would I give you a bath? Peter did a good job realizing that this was a spiritual lesson but he wasn't getting the message. Lets take a look at the passage you are having trouble with:
"*Jesus says to him, He that is washed all over needs not to wash save his feet, but is wholly clean; and ye are clean, but not all. *For he knew him that delivered him up: on account of this he said, Ye are not all clean."
(John 13:10,11, 1890 Darby Bible)
Did you notice that Judas is mentioned? Judas had a problem with vain ambition as well and it had finally destroyed him. Jesus told them that they would not understand this lesson until later but Peter refused to be taught because he thought he allready knew everything. The posters seem to have made the insight that this passage is a lesson on sanctification (the washing, renewing and regeneration of the Spirit, see Titus 3:1-6) but so did Peter. The insights are all good but I am concerned that the context of the verse in question has been neglected and that is where the confusion came from in the first place.
What was making Peter dirty, his feet or his heart? The real answer is both, but the feet problem is easily fixed, the heart however is considerably more complicated. Did you ever stop to think, Jesus washed Judas' feet and Judas walked out the door and betrayed him. At least one of them never got this lesson, Jesus said that it would better for him that he had never been born. If you want to learn the meaning of this passage, go wash someone's feet.
Now for your questions:
Is Jesus saying he is clean every whit?
I think what he is saying is that one of you was never cleaned, note the reference to Judas. It is also a kind of humorous statement that only your feet are dirty.
Was Peter not clean even after Jesus washed his feet
Peter was allready clean through the washing of the Word of God that he had accepted by faith.
"*Husbands, love your own wives, even as the Christ also loved the assembly, and has delivered himself up for it, *in order that he might sanctify it, purifying it by the washing of water by the word"
(Ephesians 5:25,26)
What does he mean when he says "He that is washed needeth not save to wash his feet?"
It means both that only Peter's feet are dirty and the even though you saved by faith the moment you recieve the Gospel, there are still sins that will try to choke out the Word of God in your heart. (See Mark 4:1-20)
Grace and peace,
Mark
Wait! I just thought of something! Remember how John said that he was not worthy to unloose the sandles of the one who would come later? That passage has baffled me but now it makes sense, John is saying that he isn't worthy to take off his sandles, let alone wash his feet. Imagine how supprised John would have been to learn that the Messiah came to wash his feet. When Jesus came to be baptised John did say that he needed to be baptised by Him.