This may not be fully in-line with what you're after, since the main suggestion is to drop a bunch of your manual formatting:

\documentclass{article}
\usepackage{relsize,amsmath}
\begin{document}
\[
g^{\mathsmaller{\left(1\right)}} \left( \tau \right) =
\frac{\langle E_{\mathsmaller{s}}^{\mathsmaller{*}}\left( 0 \right)
E_{\mathsmaller{s}} \left( \tau \right) \rangle}
{\langle \lvert E_{\mathsmaller{s}} \rvert^{\mathsmaller{2}} \rangle}
\]
\[
g^{(1)}(\tau) =
\frac{\langle E_s^*(0) E_s^{\vphantom{*}}(\tau)\rangle}
{\langle \lvert E_s \rvert^2 \rangle}
\]
\end{document}
I've removed the \mathsmaller font adjustment, as well as the over-use of \left...\right constructions. Their use typically introduces additional/unwanted space.
Specific to your points:
To correct for the placement of the mis-aligned s, insert a non-existent * using E_s^{\vphantom{*}}
\mathsmaller only reduces the font size (from \scriptstyle to \scriptscriptstyle in your case), but doesn't change the baseline. As such, it seems like the superscripts can be a little higher. If you use the regular (\scriptstyle) superscripts, they are appropriately aligned with the top of the superscripted base.
Ps. Final OCD-ance: Perhaps you could add a small space \, between the operands in the fraction numerator to distinguish them properly, leaving to this output:

s, useE_s^{\vphantom{*}}. The lower placement of*and2is due to the fact that the superscript baseline still remains the same regardless of the use of\mathsmaller. What should also tickle your OCD is the excessive use of\left...\right, which causes wide (unnecessary) spaces around them. – Werner May 22 '14 at 16:08