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I haven't read your full question, but the answer to: Is there an equivalent analytical result where we can add Gaussian noise proportional to each coordinate sensitivity? and (implicitly) Can the noise scale better than $O(d)$ for $d$-dimensional output? Then the answer is yes. The following should be able to be easily adapated for essentially any ...
Apologies for seeing this question just now. For your first question, well, there seems to be a typo: I should have written $\|s\|^2 \leq n/2$, and then we have $\sqrt{(\|s\|^2 +1) /12 } = \sqrt{251/12} \approx 4.57$. Now, we take 2 ciphertext with LWE error of std-deviation $\beta$, and sum them. Assuming independence the std-dev of the sum of the errors ...