What are the important plot complications in order, with the details and conflict of each, in Twain's The Prince and the Pauper?
This format doesn't permit for a discussion of all the plot complications, but I can give some overview and some detail. First, of course, Tom Canty gets into the palace where he and Crown Price Edward exchange clothes and discover they look identically alike. While still in each other's clothing, Edward is accosted and evicted by the gate guard when he attempts to have the gate unbarred ("Open! Unbar the gates!"). This sets up a conflict between reality and the appearance of reality. In each case the reality appeared to be a fabricated delusion. A second conflict for each lay in human versus environment as each struggled to adapt to an environment thrust upon him.
"Then so shall it be. Doff thy rags, and don these splendours, lad! It is a brief happiness, but will be not less keen for that. We will have it while we may, and change again before any come to molest."
After this, there are the complications of how each is treated and how each reacts. Tom fared better than Edward as Tom was accepted and humored by those who loved him [Edward] and sought to remedy his temporary (it was hoped) madness. On the other hand, Edward, rightly--but unfortunately--demanding his due respect, had to be rescued to remain safe.
The end result for Edward is that after being in many dangerous situations, he arrives with his rescuer Miles at the coronation of King Edward [Tom] just as he is about to be crowned. Edward successfully halts the ceremony when Tom recognizes him and is challenged by the Lord Protector to reveal the location of the missing Great Seal of State. He does and all is then resolved. The final complications that lead to the resolution further involve the conflict of human against environment. This time, however, each has learned to adapt to the environment and can acquiesce or fight accordingly, as need arises.
Tom acquiesces as he doesn't know how to contact Edward and is never out of someone's protective sight--plus he fears the horrible punishment if found out. Edward fights with will and strength to right wrongs he is not responsible for. [Edward originally left Tom alone to go out and, the text infers, speak to the ill King about the mistreatment Tom received from the guard, as evidenced by the bruise on Tom's hand.]
[At the coronation,] He raised his hand with a solemnity which ill comported with his soiled and sorry aspect, and delivered this note of warning—
"I forbid you to set the crown of England upon that forfeited head. I am the King!"
The general message of the plot complications and conflicts points to the difficulty of authentic (truthful) representation of self in an environment where one is disbelieved, discredited, and/or disrespected. [Tom would have been disbelieved and was in ways discredited; Edward suffered the extremes of each.]