Any bolded actions, or actions that use the word "may" are optional. Everything else is mandatory.
There are two actions which trigger off Frodo's death: threat wounds and Sam's action. Since both have the same trigger, the mandatory action HAS to go first. If there are two or more mandatory actions, the FP chooses which one will occur first. You will often see this kind of choice on Gemp when, say, a Dwarf with a
Dwarven Axe wins a skirmish; as the FP, you choose for either the axe's text to discard a card occurs first, or placing the wound(S) on the losing minion occurs first.
And to clear one other thing up, yes, there is a brief period of time where you actually have no Ringbearer in play while you are assigning the threat wounds due to Frodo's death.