Tell me if this is right:

It is possible mathematically to represent a countably infinite number of immortal people, as well as the process of moving them between spheres. Further, we should not expect a priori that a problem involving such infinities would have a solution equivalent to those solutions reached by taking infinite limits of an analogous finite problem.

That's an accurate interpretation of my comment.

Some confusion arises when we introduce the concept of “utility” to determine which of the two choices is better, since utility only serves as a basis on which to make decision for finite problems.

I do think that confusion arises in this context from the concept of "utility", but not because "utility only serves as a basis on which to make decision for finite problems." The "utility" in the problem is clearly not that of VNM-utility (of which I previously gave a brief explanation) because we not assigning utility to actions, decisions, or choices (a VNM-utility function U would generally have no problem responding to an infinite set of choices, as it simply says: do argmax_{choice}(U(choice))). This severely undermines what we can do with the "utility" in the problem because we are left with the various flavors of aggregative utilitarianism, which suffer from intractable problems even in finite situations! Attempting to extend them to the situation at hand is problematic (and, as Kaj_Sotala remarked, dealing with infinities in aggregative consequentialism is the topic of one of Bostrom's papers).

1. Do you view the paradox as therefore unresolvable as stated, or would you claim that a different resolution is correct?

I think that the appearance of the paradox is a consequence of unfamiliarity with infinite sets, and that it is not too surprising that our intuition appears to contradict itself in this context (by presenting each option as better than the other). The contradictory intuitions don't correspond to a logical contradiction, so the apparent paradox needs no resolution. The actual problem (choosing between the two options) is a matter of preference, just as the choice between strawberry and chocolate is a matter of preference.

2. If I carefully restricted my claim about ill-posedness to the question of which choice is better from a utilitarian sense, would you agree with it?

Absolutely. I think aggregative utilitarianism (as a moral theory) is screwed even in finite scenarios, much less infinite scenarios. (But I also think aggregative utilitarianism is a good but ill-defined general standard for comparing consequences in real life.)