\cmidrule with two multirows not behaving correctly?

For having some space between two \cmidrules at the same level, you never want \morecmidrules. Just use the () optional argument to shorten the line, as in the following:

\begin{table}
\centering
\begin{tabular}{rrrrr}\toprule
  $b$ & \multicolumn{2}{c}{QCDH} & Single & \smash[b]{$\frac{\pi_{q1}}{\pi_{s1}}$} \\
  \cmidrule(r){2-3}\cmidrule(l){4-5} % notice there's no \\ here
  & $\pi_{q1}$ & $\pi_{q1}$ & $\pi_{s1}$ & \\
  \midrule[\heavyrulewidth] % notice there's no \\ here
0.1 & 0.0055 & 0.0018 & 0.0018 & 3.0555\\ \midrule
0.5 & 0.1218 & 0.0830 & 0.0074 & 16.4594\\ \midrule
2.5 & 0.7325 & 0.6771 & 0.3561 & 2.0570\\ \bottomrule
\end{tabular}
\caption{Pickup rates from QCDH and single SNP approach}
\end{table}

Integrating the functionality of the multirow package with that of the booktabs package is tricky: \midrule and \cmidrule add extra vertical spacing, but this extra spacing isn't taken into account when a \multirow instruction is executed. Thus, lining up the material in a 2-row multirow that also features \midrule or \cmidrule between those two rows is rarely satisfactory.

Moreover, I'd say your table neither requires nor benefits from applying a \multirow to the contents of columns 1 and 5 of the header. Just place those items in the very first row, and make the \cmidrule span just columns 2 and 3.

I think it's in general a good idea to center the items in the header even if the material in the body of the table isn't (or shouldn't be) centered. Finally, since the data are numeric, it's probably also a good idea to align them on their decimal points.

\documentclass{article}
\usepackage{amsmath,booktabs,siunitx}
\begin{document}
\begin{table}
\centering
\begin{tabular}{S[table-format=1.1]
           *{3}{S[table-format=1.4]}
                S[table-format=2.4]}
\toprule
    {b} & \multicolumn{2}{c}{QCDH} & {Single} & {$\dfrac{\pi_{q1}}{\pi_{s1}}$} \\
    \cmidrule(lr){2-3}  
 & {$\pi_{q1}$} & {$\pi_{q1}$} & {$\pi_{s1}$}  \\
\midrule 
0.1 & 0.0055 & 0.0018 & 0.0018 &  3.0555\\
0.5 & 0.1218 & 0.0830 & 0.0074 & 16.4594\\
2.5 & 0.7325 & 0.6771 & 0.3561 &  2.0570\\
\bottomrule
\end{tabular}
\caption{Pickup rates from QCDH and single SNP approach}
\end{table}
\end{document}