I don't understand your question. What does it mean $I$ integrable? Everything you need is proven in the Balazard papers, why don't you follow them. $\frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}dt$ is locally integrable (because $\zeta$ is meromorphic), if it is integrable then $I=\lim_{\epsilon\to 0}I_\epsilon$ trivially. Is it integrable?

Did you read my comment?

your question are not well-posed (hint $\liminf (t_{n+1}-t_n) = 0$, so for fixed $\epsilon$ there is a large $n(\epsilon)$ for which $[t_n+\epsilon,t_{n+1}-\epsilon]$ becomes ill defined at least for some $n$ and I hoped after all this discussion you would get that, how to modify them to be well posed (one way is to define $I_{\epsilon}$ only for $n < n(\epsilon)$) and how to answer them (local integrability to ensure $I_{\epsilon}$ exists and global integrability to insure the limit exists;

do you know that $\liminf (t_{n+1}-t_n) = 0$?

because it means that if you fix $\epsilon >0$, the interval $[t_n+\epsilon,t_{n+1}-\epsilon]$ becomes ill defined for infinitely many large $n$

beacuse $(t_{n+1}-t_n) < \epsilon$ for infinitely many large $n$

not sure about the goal of the question; the way one solves this (usually) is to restrict to $n(\epsilon)$ for which $(t_{n+1}-t_n) < \epsilon, n \le n(\epsilon)$

do you know how many zeroes (unconditionally proven by Selberg and improved by others) does RZ have on the critical line up to level $T$?

of course but that is general theory of holomorphic functions; for RZ much more precise results are known

correct - which means that the average spacing of the zeroes is what?

let's spell it out (a rough heuristic argument that can be made rigorous); up to $T$ we have $cT\log T$ zeroes, and up to $T+1$ we have $c(T+1)\log (T+1)$ zeroes, so how many roughly are between $T$ and $T+1$?

The answer is $c(T+1)\log (T+1)-cT \log T=c\log (T+1)+O(1/T)$ hence the average spacing is $c/\log T$ which is also (roughly) $c/\log t_n \to 0$ so the spacings go to zero at least for a subsequence, hence the OP is ill formulated as it stands;

$t_n+\epsilon >t_{n+1}-\epsilon$

not sure of what you want to prove in the post

$[0, t_1-1/2]$ works where $t_1=14...$ is the first zero ordinate; not sure why does that help here

$[t_n+\epsilon_n,t_{n+1}-\epsilon_n]$ for any $\epsilon_n < \frac{t_{n+1}-t_n}{2}$ (in other words, there is no fixed $\epsilon$ that works for all $n$

yes that would work; but note that $\liminf \epsilon_n=0$

what limit? there are infinitely many $\epsilon_n$ so what does it mean to take a limit on them? as noted to prove that $I$ exists and is finite is trivial and doesn't require all these convolutions (truncate it at $N$, show that the remainder is negligible etc - ); the saddle part is to show the Balazard et others closed form and that is also not that deep btw

what does that mean? there is no $I_{\epsilon_n}$ but an $I_{(\epsilon_1, \epsilon_2,...\epsilon_n,...})$

if my understanding is correct you want to excise disjoint intervals around critical zeroes and their lengths depend on $n$; so in other words, the integral obtained would depend on an infinite sequence of different lengths and how you index it is up to you but then it becomes the question of what do you mean by taking a limit (as already at least a subsequence of the lengths go to zero with $n$ for a Fixed choice of the excisions); again not sure why try something like this when stopping at a finite $N$ works in defining $I$ and proving its finiteness and is the customary way done

have no idea what the question above means; as noted I wouldn't approach $I$ this way since I am not sure how to make sense of a limiting process on infinite sequence

have no idea as noted since I do not know (and have no interest to think about) how to make sense of a limiting process of an infinite sequence of an infinite sequence (at least here where other techniques are available to deal with $I$)

this last part was definitely more interesting and while not that deep, definitely more RZ like good question; local integrability at zeroes and estimates at infinity on the critical line are fairly trivial; still not clear about the purpose of the question