What does it mean to be a Prince of Amber?
Princes have a mixture of rights and privileges which others are denied, though some are shared by any spouses they may have.
Princes are bound to defend the Patterns and obey the crown of Amber. (Or Ember or Arcadia or Rebma, etc.) Your blood can destroy it; you must be willing to spend your blood to defend it, in return for the power it has granted you. And every prince bows to the Pattern Kings and Queens. (Except, perhaps, a few who were raised in strange places and have yet to choose a fealty.)
Princes can commit no crime but treason. The Code of Duels and their obedience to their monarch binds them, but to a large extent, a Prince can do no wrong so long as they avoid treason.
That being said, other Princes may take offense at their actions, and challenge them to a Rose Duel. Certain others have been granted Rose Rights as well. The person challenged may choose the weapon to be used, and may choose a champion, if they feel they cannot afford to back down, but themselves have no hope of defeating the challenger. Should the challenger choose some form of combat impossible or extremely disadvantageous to the challenger, they may themselves find a champion if anyone will stand with them.
Some such duels are simply an excuse for beating each other up, but usually some sort of conditions are set which the loser must grant to the winner.
Should the challenged refuse a duel entirely, the Prince may appeal to the King and Queen to grant them the right of Vendetta. This allows violent action to be taken against others normally protected by the Code of Duels. Whether this will be granted typically depends on how severe the offense was. Should it be refused, it may still be possible to bring suit in the royal courts to force some sort of repayment/punishment.
(It is to be noted that assaulting or killing a Prince or their spouse without grant of Vendetta right is within the definition of treason unless it be in self-defense or defense of one's spouse or another Prince.)
Princes cannot resign their royal status as a Prince of Amber/Rebma/Whatever, nor can they be stripped of this status. Other offices and titles (Duke of Earl, Lord High Executioner, King of Amber) can be resigned or taken away, but you are a Prince until you die, like it or not.
Extremely High Treason:
Trying to destroy any of the Elemental Patterns or the Primal Pattern is Extremely High Treason. You will know pain and know it abundantly, and then you will die.
An assault with intent to kill on the King, Queen, or any Prince is High Treason. High Treason is always tried by the King or Queen and carries the death penalty (or possibly life imprisonment) for anyone.
Abuse of High Office is also High Treason.
Making war on Amber, or providing aid and comfort to its enemies is also High Treason.
Low Treason is an assault on any royal officer. The abuse of office by lower ranking royal employees is low treason. Counterfeiting and Forgery of royal documents are also Low Treason.
Non-Fatal assaults by Prince against Prince theoretically fall into this category.
Low Treason brings the Death penalty for non-Princes, and can be punished with long terms of imprisonment for Princes.
It is theoretically illegal for Princes to randomly beat the crap out of each other for being stupid. In practical parlance, Tylor typically declares that you effectively consented to a duel by giving provocation for beatings.