These seem fairly advanced, so I don’t know if I would call them “common”. But maybe I am out of the loop. Are these concepts and very specific sounding beliefs actually widely known to math majors or CS majors?
To the question of how advanced these are, with only a few exceptions, the majority of things there are covered in undergraduate analysis, algebra, and topology courses.
At the University of Waterloo, CS is strictly in the faculty of Math, so I'd expect students to know much of this (I think? I'm in Computer Engineering at UW, which is a distinctively different program).
OTOH, I'd imagine things are different in universities where CS is thought of as its own major, with no Software Engineering/Computer Engineering as an alternative.
Things like "Many students believe that 1 plus the product of the first primes is always a prime number" (a misunderstandment of Euclid's proof that there exist infinitely many prime numbers) or "If A implies B then B implies A" (difference between a necessary and a sufficient condition) are literally middle-school math.
Things like "If 𝑓(𝑥,𝑦) is a polynomial with real coefficients, then the image of 𝑓 is a closed subset of ℝ" are high-school level.
The top answer about dimensions of vector spaces is a well-known undergraduate-level "trap".
"The Krull dimension of a noetherian integral domain is finite." is, to be brutally honest, not a sentence I understand in the slightest.
Also
> "If 𝑓(𝑥,𝑦) is a polynomial with real coefficients, then the image of 𝑓 is a closed subset of ℝ"
I don't think that this is high school level. Secondary school level math rarely touches anything related to multivariate polynomials (and usually doesn't have a notion of "closed sets" beyond closed intervals, but that's not as relevant apart from how the problem is stated).
I don't doubt that what you're saying is true, it's just something far outside what I personally am familiar with.