How about that, Bib and I agree.
Frodo dying starts a new chain of actions. Anything that triggers off the death of Frodo happens at that time, including Sam taking the ring. Sam can take that response even though it's not a required action, because he's keying off the death of Frodo (new action chain), not the wounds themselves.
dmaz, look at how things like
Deep in Thought work. Once again, it's causing a bunch of effects at once, but they still happen one-at-a-time. That's how
Deceit is able to prevent cards from being discarded.
Also, it just occurs to me that we may be mixing up "required actions" with "effects." The original question was in regards to
Berserk Slayer's text. That text is an effect, not a required action. A required action would be something more like
Minas Tirith First Circle.
Bib is right about the threat timing, because threat wounds are another thing that key off the death of a character. Because they're a required action, you'd have to place the threat wounds before you take the Response action of Sam taking the ring.
You've already quoted the pertinent sections of the rules yourself.