Is the the spectral norm of a Boolean function bounded by the degree of its Fourier expansion?
It is a standard fact that if $f:\{-,1,1\}^n \to \{-1,1\}$ is a function of Fourier degree $d$, then its Fourier coefficients are multiples of $2^{-d+1}$. In particular, every non-zero coefficient must be at least $2^{-d+1}$ in absolute value. Therefore, by Parseval, there are at most $2^{2(d-1)}$ non-zero coefficients, and so the spectral norm of $f$ is at most
$$\sum_{S}|\hat{f}(S)| \leq \sqrt{2^{2(d-1)}}\sqrt{\sum_{S}\hat{f}(S)^2} = 2^{d-1}$$.

This bound is tight. For example the complete binary decision tree of depth $d$ has spectral norm $2^{d-1}$. This can be shown, e.g., by induction on $d$.
The address function has also maximal possible spectral norm.