"Whether intentionally or not, John always seems to be out “front, running” ahead of the pack and “blazing the trail.”
The above notions kind of imply “firstness,” and derived from them there are: “Trailblazer” and “Front-runner” for single-word nouns.
Trailblazer: “a person who is the first to do or discover something
and so makes it possible for others to follow”
(from Oxford Learner’s Dictionary)
Front-runner or frontrunner is a term used to describe the leaders in
a race, whether political or athletic.
Although “trailblazer” and “front-runner” might capture in one word the notion of “a person who is first,” neither of them capture the notion or characteristic of “firstness” that you are seeking.
Combining them, however, with something like “spirit,” “nature,” or “tendencies” to get noun phrases like “trailblazing/trailblazer spirit/nature/tendencies” and “front-running/front-runner spirit/nature/tendencies” could work in your first two examples with slight modification:
“Roger Bannister was the first person to run a mile in under four minutes. Not many of us have the front-running/front-runner spirit/nature/tendencies needed to achieve [what he achieved].”
“Neil Armstrong exhibited [his] trailblazing/trailblazer spirit/nature/tendencies by [doing what he did].”
Your third example would require creating a phrase with another word in order to make “front-running” work, like “reckless,” “compulsive,” or even “pointless”:
“Some people prefer to answer first in a quiz, even if they get the answer wrong. They prefer reckless/compulsive/pointless front-running over correctness.”
The only single word that I can see that could possibly work in all three examples would be “pacesetter/pacesetting,” and this only because it has two meanings:
… [related forms: pace′set′ting adj. & n.] …
a person or group that serves as a model to be imitated or followed; leader.
one that sets the pace, as in racing.
(from ‘American Heritage® Dictionary of the English Language, Fifth Edition.’ Copyright © 2011 and ‘Random House Kernerman Webster's College Dictionary,’ © 2010 both via ‘The Free Dictionary’)
“Roger Bannister was the first person to run a mile in under four minutes. Not many of us can achieve pacesetting [pacesetter/setting status]/[be pacesetters] in something.” (definition #1)
“Neil Armstrong exhibited pacesetting/[is/was/became a pacesetter] by being the first person on the moon.” (definition #1)
“Some people prefer to answer first in a quiz, even if they get the answer wrong. They prefer pacesetting over correctness.” (definition #2)