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-- Eragon and Angela, Brisingr, by the same author
Someone who says something like the first sentence generally means something like "questions that are significant and in an area I am concerned with". They don't mean "I don't know exactly how many atoms are in the moon, and I find that painful" (unless they have severe OCD based around the moon), and to interpret it that way is to deliberately misinterpret what the speaker is saying so that you can sound profound.
But then, I've been on the Internet. This sort of thing is an endemic problem on the Internet, except that it's not always clear how much is deliberate misinterpretation and how much is people who just don't comprehend context and implication.
(Notice how I've had to add qualifiers like 'generally' and "except for (unlikely case)" just for preemptive defense against that sort of thing.)
If you don't have any open questions in that category, then you aren't really living as an intellectual.
In science questions are like a hydra. After solving a scientific problem you often have more questions than you had when you started.
Schwartz's article on the issue is quite illustrative. If you can't deal with the emotional effects that come with looking at an open question and having it open for months and years you can't do science.
You won't contribute anything to the scientific world of ideas if you can only manage to concerned with an open question for an hour and not for months and years. Of course there are plenty person in the real world who don't face questions with curiosity but who in pain when dealing with them. To me that seems like a dull life to live. because the question doesn't concern themselves with living an intellectual life.
I'm not sure that's a critical part of any definition of the word "intellectual".
It's not sufficient to be an intellectual but if you don't care about questions that aren't solved in short amounts of time because that's very uncomfortable for you, you won't have a deep understanding of anything. You might memorise the teacher password in many domains but that's not what being an intellectual is about.
"You must spend every waking hour in mortal agony, for life is full of unanswerable questions." carries the connotation that someone cannot answer large numbers of every day questions, not that they can't answer a few questions in specialized areas.
But the original statement about unanswered questions being painful, in context, does connote that they are referring to a few questions in specialized areas.
In this case it illustrates how the character in question couldn't really imagine living a life without unanswered questions. Given that it's a Science Elf that fits.
For him daily life is about deep questions.
"Unanswered questions" connotes different things in the two different places, though. In one place it connotes "all unanswered questions of whatever kind" and in another it connotes "important unanswered questions". The "cleverness" of the quote relies on confusing the two.
Important depends on whether you care about something. If you have a scientific mindset than you care about a lot of questions and want answers for them.
But you don't care about the huge number of questions needed to make the response on target.
I just liked seeing the usually-untouchable hero called out on his completely empty boast of how tirelessly curious and inquiring he was.