What you have is positive integers such that $\;a_1b_1+\dots+a_sb_s=24.$ Your $q/\eta_\sigma(q)$ is an example of an eta-quotient and is a modular function of negative weight. As just one example, if $\;a_1=1,b_1=24\;$ then $\eta_\sigma(q)=\Delta(q)$ is the generating function of the Ramanujan tau function. You may find some similar kinds of partitions in the OEIS. For example, sequence A005758 is partitions into parts of 12 kinds. For your first question, just expand $\;\eta(q^a)=\prod_{n>0}1-q^{an}\;$ and multiply the $q$-series together.

What you have is positive integers such that $\;a_1b_1+\dots+a_sb_s=24.$ Your $q/\eta_\sigma(q)$ is an example of an eta-quotient and is a modular function of negative weight. As just one example, if $\;a_1=1,b_1=24\;$ then $\eta_\sigma(q)=\Delta(q)$ is the generating function of the Ramanujan tau function. You may find some similar kinds of partitions in the OEIS. For example, sequence A005758 is partitions into parts of 12 kinds. For your first question, just expand $\;\eta(q^a)=\prod_{n>0}1-q^{an}\;$ and multiply the $q$-series together. For example, in your case where $b_1=b_2=a_1=1, a_2=23,$ $$f(q):=\frac1{\prod_{k>0}(1-q^k)(1-q^{23k})}=1+q+2q^2+3q^3+5q^4+\dots$$ which is the generating function of partitions of $n$ into positive integers where the integers come in two kinds and the second kind has weight $23$ times the weight of the first kind.