Rather than updating the question, let me devote a separate answer to discuss the emerging knowledge. (Please corrent me if I make any mistake.) I will update this answer when necessary.
First, the Prime Number Theorem in its stronger known form asserts that
$$|\sum_{k=1}^X \mu(k)| \le e^{\sqrt \log X}. $$
And the RH asserts that $$|\sum_{k=1}^X \mu(k)| \le X^{1/2+\epsilon}.$$
Let $X=2^n$, the Prime number theorem deals with the Walsh coefficient
$\hat \mu(\emptyset)$.
(Remark: I am still a little confused about the situation, since the upper bound for the ordinary discrete Fourier coefficients in this answer by Matt Young are not as strong as the statement for the 0th coefficient given by the PNT.PNT. This is now clarified by Ben's remark below.)
The first question
The first question was if $\hat \mu(S)$ tends to zero uniformly with X and at what rate.
A special case of interest was the correlation between the Mobius function and the Morse function (which is \hat \mu ([n])$. Ben Green noted that the method by Mauduit and Rivat gives directly that
$$\hat \mu ([n]) \le X^{-c}, $$
for some positive constant $c$.
Also according to Ben the results and methods of Harman and Kátai will give that $\hat \mu (S)$ uniformly tends to zero whenever $|S| \le n^{1/2-\epsilon}$ (in fact they give a stronger result mentioned below).
According to Ben, the technique of Mauduit-Rivat are likely to work unless S∩[n/3] is very ``thin'', and with more effort combining Mauduit-Rivat and Herman-Katai.
The second question and the $AC^0$-prime number conjecture.
Ben wrote that the Herman-Katai method ought to give that
$$\hat \mu (S) \le e^{\sqrt \log X}, $$ whenever $|S| \le n^{1/2-\epsilon}$.
This is more than enough to imply a positive answer to question 2.
(At present an argument is sketched under GRH.)
Ben's positive answer to question 2 implies the $AC^0$- Prime Number Conjecture (a.k.a Sarnak-Kalai conjecture)! The implication is rather direct. Hastad Switching lemma implies (as noted by Hastad and by Boppana) that the total influence (a.k.a. average sensitivity) of an $AC^0$ Boolean function is polynomial in (log n) and this implies that most of the Fourier-Walsh coefficients are below the polylog level which together with an affarmative answer to question 2 gives the $AC^0$ PNC. There is an even stronger result by Linial-Mansour-Nisan (which was improved a little by Hastad) that asserts that the Fourier-Walsh coefficients decay exponentially above their expected value.
The relation with known CS literature?
In the question there are links to several papers which deals with related question of the inability of $AC^0$ functions to compute certain number theoretic questions. These papers rely heavily on Fourier expansion of $AC^0$ circuits, Linial-Mansour-Nisan, Hastad etc.
I still have to exlore these papers more to see if they discuss explicitely or implicitely upper bound for Fourier-Walsh coefficients. It seems that the paper by Anna Bernasconi and Igor Shparlinski and some papers cited there are most relevant. It looks that there is a proof there that there is much weight of the Fourier coefficients of a function expressing square-freeness (which is very close to Mobius) is on coefficients with large value of |S|. But I am not sure if Bernasconi-Shparlinski bounds already give the $AC^0$ PNC. (Also the arguments look different.)
Follow up questions
1) Give an affarmative answer to question (1)
2) Extend the PNT when you consider functions in ACC[p] namely you allow addition mod p gates. Note that question (1) is a very special case of ACC2, It would be nice to "merge" the Rasborov-Smolensky method to deal with ACC[p] fubctions with some ANT. Now that Ben settled the PNC for $AC^0$ functions this will be a natural next step.
3) Give an affermative answer to question 3. It will imply that under GRH the AC^0 PNT extends "almost" all the way to log-depth.
Showing that $\hat \mu (S) \le X^{-1/3}$ will imply "The prime number conjecture for monotone Boolean functions" namely that the Mobius function is asymptotically orthogonal to every function described by a monotone Boolean function of the digits. (No complexity assumptions.)
(This probably implies statement like: if you consider a randon sequance of integers $0=X_1,...,X_n$ so that $X_{i+1}$ is obtained from $X_i$ by switching a digit from 0 to 1 then the Mobius function will change sign many times on the sequence.)
4) It would be interesting to see if appropriate low level complexity classes (Also allowing random inputs to the circuits) account for other known results about "Mobius randomness". Interesting examples: Standard L functions, the Green-Tao bracketed polynomials; non deterministic sequences in the sense of Peter Sarnak,
