Compute $\pi^n(S^1\times S^{n+1})$. What is the space of homotopy classes of maps $S^1\times S^{n+1}\to S^n$? Is there a simple way to compute it, if we know $[S^{n+1}, S^n]\simeq\mathbb{Z}^2$ (resp. $\mathbb{Z}$ for $n=2$)?

Here's an attempt assuming you are interested in unbased homotopy classes of maps.
Let $[X,Y]$ denote based homotopy classes of maps, then what we are looking for is the space $[S^1_+ \wedge S_+^{n+1},S^{n}]$
\begin{align}
[S^1_+ \wedge S_+^{n+1},S^{n}] &= [S^{n+1}_+,Maps(S^1_+,S^n)] 
\end{align}
$Maps(S^1_+,S^n)$ is the free loop space $LS^n$.
We have a split fibration $\Omega S^n \rightarrow LS^n \rightarrow S^n$, the splitting is via inclusion of $S^n$ in $LS^n$ as constant loops.
When $n>2$, $LS^n$ is simply connected and we have 
\begin{align*}
[S^{n+1}_+,LS^n] &= [S^{n+1},LS^n]\\
&=\pi_{n+1}(LS^n)\\
&=\pi_{n+1}(\Omega S^n) \oplus \pi_{n+1}(S^n)\\
&=\pi_{n+2}(S^n) \oplus \pi_{n+1}(S^n)
\end{align*}
For $n=1$ we have $$[S^2_+,LS^1] = [S^2_+,S^1 \times \mathbb{Z}] = \mathbb{Z} \oplus \mathbb{Z} $$
(see https://mathoverflow.net/a/149664/29548)
Not sure what happens for $n=2$.
