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It seems to me that Zeno's paradoxes leverage incorrect, naïve notions of time and computation. We exist in the world, and we might suppose that that the world is being computed in some way. If time is continuous, then the computer might need to do some pretty weird things to determine our location at an infinite number of intermediate times. However, even if that were the case, we would never notice it – we exist within time and we would not observe the external behavior of the system which is computing us, nor its runtime.
Don't have much of an opinion - I haven't rigorously studied infinitesimals yet. I usually just think of infinite / infinitely small quantities as being produced by limiting processes. For example, the intersection of all the ϵ-balls around a real number is just that number (under the standard topology), which set has 0 measure and is, in a sense, "infinitely small".