Are there some general rules/formulas on the relation between the homotopy class $[f]\in \pi_i(S^n)$ and the homotopy class of the composition $S^i\stackrel{a}{\to} S^i\stackrel{f}{\to}S^n\stackrel{b}{\to}S^n$ where $a,b$ are maps of degree $d_a,d_b$ respectively? I think that composition with $a$ always multiplies $[f]$ by $d_a$, but composition with $b$ seems to be harder. Is $[b\circ f]$ always a multiple of $[f]$?
Apparently, if $f: S^3\to S^2$ is the Hopf map, the homotopy class of the composition is $(d_a \times d_b^2) [f]$ (see this book, p. 205). In the stable dimension range, however, composition with $b$ seems only to multiply $[f]$ by $d_b$, if I understand this wikipedia paragraph correctly (supercommutativity).
(this is not a full answer, but for comments it's too long)
For given (smooth) map $f:S^3\to S^2$ if you take two non-critical values $p,q\in S^2$, then linking number of $f^{-1}(p)$ with $f^{-1}(q)$ equals to degree of $f$. Maybe, something similar occurs in case $i=2n-1$ for all $n$.
And when you take a suspension of the diagrams $S^i\stackrel{a}{\to} S^i\stackrel{f}{\to}S^n\stackrel{b}{\to}S^n$, for stable dimensions $i,n$ answer will be the same, because of map $\pi_i(S^n)\to\pi_{i+1}(S^{n+1})$ being surjective and equalities $d_{\Sigma a}=d_a$, $d_{\Sigma b}=d_b$.