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Solved in the positive by Jean Bourgain (April 12, 2011): Moebius-Walsh correlation bounds and an estimate of Mauduit and Rivat; (Dec, 2011) For even stronger results see Bourgain's paperOn the Fourier-Walsh Spectrum on the Moebius Function.

(This does not seem to follow from bounds we can expect unconditionally on individual coefficients.) Solved in the positive by Ben Green (March 12, 2011): On (not) computing the Mobius function using bounded depth circuits. (See Green's answer below.)

Question 3 suggestes that perhaps we can deduce the $AC^0$ prime number conjecture from the GRH which would be of interest. Of course, it will be best to prove it unconditionally. (Ben Green proved it unconditionally).

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