The first Pontrjagin class of Milnor's example of exotic 7-sphere

1. This is a general fact. If $G$ is a topological group and $f\colon S^{n+1}\rightarrow BG$ classifies a principal $G$-bundle $p\colon E\rightarrow S^{n+1}$, then the naturality of the fibration LES tells you that the image of $[f]$ under $\pi_{n+1}(BG)\stackrel{\sim}{\rightarrow}\pi_n(G)$ takes $[f]$ to the image of $\mathrm{id}_{S^{n+1}}$ under the boundary map $\partial\colon\pi_{n+1}(S^{n+1})\rightarrow\pi_n(G)$ for the fibration $p$. Now, choose trivializations $\varphi_{\pm}\colon E\vert_{D^n_{\pm}}\rightarrow D^n_{\pm}\times G$ over the hemispheres and let $\psi\colon S^n\rightarrow G$ be the induced clutching map. Then, $S^n\times [-1,1]\rightarrow E,\,(x,t)\mapsto\varphi_-^{-1}(\pi(x,t),e)$ for $t\le0$ and $\varphi_+^{-1}(\pi(x,t),\psi(x))$ for $t\ge0$ define a lift of the quotient map $\pi\colon S^n\times[-1,1]\rightarrow S^{n+1}$ that is constant on $S^n\times\{-1\}$, so its restriction to $S^n\times\{1\}$ defines $\partial[\mathrm{id}_{S^{n+1}}]$, but this is the clutching map $\psi$ by construction (I'm taking a cavalier attitude towards basepoints, forgive me).
2. The definition implies that $f_{i+i^{\prime},j+j^{\prime}}=f_{i^{\prime},j^{\prime}}\cdot f_{ij}$ (this means pointwise matrix multiplication), but the Eckmann-Hilton argument tells you that the pointwise group multiplication of $SO(4)$ does in fact induce the usual addition on $\pi_3(SO(4))$.

You could argue that the bundles $E_{[f]}, E_{[g]} \to S^4$ defined by $[f]$ and $[g] \in \pi_3(SO(4))$ are related to $E_{[f]+[g]}$ as follows. Consider the map $\pi : S^4 \to S^4 \vee S^4$ that crushes an equator to a point. Define a bundle $E' \to S^4 \vee S^4$ by sticking together $E_{[f]}$ over one wedge-summand with $E_{[g]}$ over the other. In other words, $E'$ is made by clutching along an $S^3 \vee S^3$ by $f \vee g$. The pre-image of that $S^3 \vee S^3$ is an equator in the original $S^4$, and $f \vee g$ pulls back to a representative of $[f]+[g]$, and therefore $\pi^* E' = E_{[f]+[g]}$. That implies $p_1(E_{[f]}) + p_1(E_{[g]}) = p_1(E_{[f]+[g]})$.

As for $[f_{i+i',j+j'}] = [f_{ij}] + [f_{i'j'}] \in \pi_3(SO(4))$, first note that the map $h_i : S^3 \to S^3, u \mapsto u^i$ has degree $i$ (every point other than $\pm 1$ has $i$ pre-images). Since elements of $\pi_n(S^n)$ are classified by their degree, it follows that $[h_i] + [h_j] = [h_{i+j}]$.

P.S. After posting this, I realise that essentially the same question was answered in "Characteristic classes of sphere bundles over spheres in terms of clutching functions"