That Cranmer is the only character to survive his trial is a puzzle. He doesn't seem to be significantly any worse or any better than the other characters who have met their demise, and he suffers from the same kind of negative rumor mill that succeeds in bringing down the other characters. Yet the king wants to save him. It is not clear what could be different about Cranmer, except that the cycle of lords blaming each other in the court had to stop at some point. But actual history reveals that it did not stop: Cranmer was saved during the span of this play, but the lord who disliked him eventually did manage to get him executed.
One of the most important parts of Henry's reign is his break with Rome. While the events that brought this about are alluded to (including the divorce from Katharine and the marriage to Anne), we see little, beyond minor references, of the kind of religious debates that must have been flourishing at the time and which led Henry to decide to break with Rome and name himself the head of the English Church.
Common people, as opposed to Lords, occupy an interesting role in the play. They are eager to follow the events of the court, from Buckingham's trial to Anne's coronation to the baptism of the baby. This, and the king and queen's enthusiasm to lower recent tax hikes, shows a kind of interclass unity unusual in Shakespeare's world. But more importantly, while the commoners on stage anxiously attend these royal events, the commoners in the audience of the play are also witness to these machinations of the courtly world on the theater's stage. Seeing the king in his court and council reduces the mystique of the monarchy and brings it to the people, making the audience into judges of the morality and actions of the royal court. Thus, a group that was largely politically disenfranchised gains some imaginary power of being able to judge the king.