Pilate and Jesus

Where does Pilate question Jesus?
Inside the Praetorium, away from the Jews, who are just outside its gate (see Praetorium for why they do not wish to enter).

How serious was Pilate when he asked Jesus, "Are You the King of the Jews" (John 18:33)?
If Pilate really thought Jesus is the king of the Jews, he wouldn't have lent one-tenth of his legion to the Jews (see Judas Iscariot) just a few hours earlier to arrest Jesus. Pilate probably posed the question to Jesus with at least some incredulity, if not sarcasm at a man who was dressed nothing like a king.

Then why did Pilate pose the question at all to Jesus?
Pilate was holding a trial and giving the defendant an opportunity to defend Himself against the charges His accusers have brought against Him.

How does Jesus reply to Pilates question?
Instead of defending Himself, Jesus responds with a question that implies He is indeed a king and turns the tables on Pilate by questioning if he recognizes this himself or is simply relaying what he has heard from others.

What does Pilate mean by, "Am I a Jew? Your own nation and the chief priests have delivered You to me" (John 18:35)?
'I am a Roman and the governor, not some Jew, and I am only relaying the charges against you.'

What does Pilate mean by "What have You done" (John 18:35)?
'You're obviously not a king, so what are you really guilty of?'

What does Pilate already appear to have ruled out?
That Jesus is some rebel leader who declared Himself king to lead an armed rebellion against the Roman rule.

What must the Roman officers who led the six hundred Roman soldiers to arrest Jesus have reported to Pilate after returning to their barracks?
That there had been no battle, no loss of life, and that Jesus had come forward and voluntarily and repeatedly identified Himself (see Jesus of Nazareth), and let Himself be arrested without a struggle (see Malchus).