Does the Hamlet Fortinbras describes (5.2.339-344) sound like the Hamlet we have known? What will happen to the kingdom under Fortinbras? Does Fortinbras get his revenge?
In my edition of Shakespeare's Hamlet, the line numbers you give feature Hamlet speaking before he dies. In my edition, I think the lines you ask about are 380-388, in which Fortinbras says:
...Let four captains
Bear Hamlet like a soldier to the stage,
For he was likely, had he been put on,
To have proved most royal,...
Essentially, all Fortinbras says is that if Hamlet had been crowned king, he likely would have proved "most royal."
What being royal consists of is probably a matter of opinion, but in general, Hamlet, throughout the play, is noble, is fully aware of the fact that actions have consequences, shows leadership abilities, apparently is an excellent swordsman, and is a good man, at least for the most part and when it comes to the larger issues of a kingdom. These traits suggest Fortinbras is correct, and Hamlet would have been most royal.
Concerning what happens in Denmark with Fortinbras in control, any answer I give would be purely specuation and irrelevant to the play. Finally, Fortinbras doesn't really have any revenge to get, since King Hamlet, the leader of the forces in the battle during which King Norway was killed, died long ago.
Unlike Hamlet and Laertes (who die getting personal revenge), Fortinbras succeeds in obtaining political revenge against the entire state of Denmark, his intentions all along. His revenge is indirect, however, and it comes almost by default since all the characters are dead upon his arrival. It is a bittersweet "bloodless coup" (you and your classmates could have easily done the same), so he doesn't get much satisfaction from storming the castle, but it is a type of revenge nonetheless.
Fortinbras is a doppelganger/foil for both Hamlet and Laertes: an amalgam of both. He has the passionate, active fervor of Laertes and the cerebral, ethical cunning of Hamlet. I think Shakespeare implies that is the ideal ruler: a balance of the Classical and modern man, one part Machiavellian and one part philosopher-king.