The particular paradoxes that you want to tie up together do seem to share an aspect from a given logical point of view.
From a 'hard Intuitionist' point of view, our linguistic construct of negation is incomplete. Brouwer traces them to our difficulty combining two different kinds of distinction related to time. And they don't fit together properly.
One is related to counting, time's successiveness, and another is related to continuity, time's 'even flow'. From the one related to counting we get the Law of the Excluded Middle, everything is either before or after this point. And from the one related to continuity we get the notion of completeness, that you can always look closer and see more stuff, but what you see won't contradict what you have already seen from a less complete viewpoint. These inherently contradict, but we can't see that clearly, and we see paradoxes when we have to choose just one of them.
Zeno's paradox is the clearest example of how they fail to fit together, we count off subdivisions of space, as our intuition of space naturally fills up all the gaps, and we are left wondering whether the process should or should not be allowed to finish. The problems with the unrestrained use of infinitesimals are just Zeno's Paradox generalized. We have to accept continuity over iteration, and that does not sit well with us. Cantor gives us some idea why. But we can't really keep it straight in our heads.
Russell's paradox is more about looking outside than inside, but from an intuitionistic point of view, the fact we can always find a set to wrap around any given set is splitting infinity in the same way Zeno is splitting a finite distance. It creates a boundlessly multiplying world, which like the continuum, 'presumably fills in' all gaps made by any distinction. We then attempt to apply the Law of the Excluded Middle to every distinction, all at once. And we fail.
Our basic notion of implication also intrinsically involves negation. We like the notion that A -> B <=> B v ~A. But this notion leads us into paradoxes like "If this statement is true, then Santa Claus said it first." Our wish to have LEM conflicts with our wish to have every statement be true or false, and this one is either both, or vacuous.
But negation is not the only source of confusion in classical logic. There are other equally weak concepts that incompletely capture other intuitions.
Our grasp on what can and cannot be compared, and why, is somewhat weak. We have a bias toward the idea that any partial order easily yields a total order. This makes us imagine the Prisoner's Dilemma should have the wrong answer, and to balk at the Game of Intransitive Dice and the related oddness in the way probabilities behave.
We can see how the first boring number would not be boring, because it would elucidate the nature of boringness, which would make it less boring. We have a definite partial order which cannot base a total order. And that is annoying.
Infinity makes this situation worse, netting us various ethical problems which involve multiple lives and attempts to compare different losses of life, which basically challenges our intuition to come up with consistent extra rules to order ratios of infinities. And the Axiom of Choice, which is equivalent to this notion that order is either there or not, via Zorn's Lemma, leads straight into the Banach-Tarski paradox about measurability, size and the nature of solidity.
Putting our problems with probability and divisibility together, statistics, which is infinitely divisible probability, creates its own host of paradoxes.
So I would claim there may be large classes of paradoxes traceable to any one specific weak intuition, but that there are at least a few unrelated sources of paradox, and beyond that different paradoxes arise by combining different sets of weak intuitions.