The quiz thing is mostly exhaustion at the idea of people who don't bother to learn anything having a say. It's no different from having kids deciding what we should have for dinner.
We can come up with a bunch of facts beforehand that everyone can read the answers to, and then it's up to people themselves to decide how much they really care.
I hear you, I feel that frustration as well. Myself, I've come the long way around to deciding that, those people need to be included anyway. And a robust system of democracy should work even with their input. Maybe at the lowest level, all voters get a "I'm happy"/"I'm unhappy" button, and signals of greater interest unlock more options and details. But I think gatekeeping is fundamentally undemocratic, and if you are going to make it harder to vote you need a better reason than "they don't seem to care".
We want more hands on the wheel, not fewer. If they don't seem to know or care, that's our problem, not theirs.
We won't know who got marked down for not answering correctly "Who was our first president". It might be busy professionals who don't have time to study (look at the CFA exam). It might be underprivileged minorities, which is a legit concern. It might be that it's evenly spread out who bothers and who doesn't.
Isn't that the point? At least from my POV, it's completely meaningless unless literally everybody is included. So yes, I'd include children, the mentally-infirm, and definitely convicts.
Think about how "universal"[0] enfranchisement came about: it has been a series of grants of influence/power, in layers, starting at the top. Barons demanded more say, and less arbitrary treatment, for themselves. They didn't get it, until they used force. Women were treated as we now treat terrorists, when they demanded a fair say.
The whole purpose of electoral systems is to deter people from trying to actually take power. So not including what you call "these people"[1] is a serious mistake.
[0] It's not universal, anywhere. Of course.
[1] The phrase "these people" is often used when othering a group of people. For example, it seems to spring easily to the lips of some Conservative politicans in the UK, when discussing immigration. It's sort of a trigger phrase for me.
I'm generally okay with criteria that are truly evenly instituted, such as age limits. I'm also okay with denying genuine transients the right to vote. These can become problematic at points, so I believe in limiting the limits. But you don't have to include literally everyone.
I would generally include convicts, and especially ex-felons, but I would never include anyone who has maliciously denied someone else the right to exercise the franchise (e.g. murderers, some kidnappers).
And some kinds of politician, I guess.
There are still issues with this, but it was an interesting idea to ponder.)