As for the second part, I think something like this works. The goal, inequality (3.4), can be written as $\newcommand{\E}{\mathbb E}$
$$
|\E[\phi_1(X)\phi_2(Y)\phi_3(Z)]-\phi_1(\E X)\phi_2(\E Y)\phi_3(\E Z)| \le C\big(1+|\sum x_i|+|\sum y_i| + |\sum z_i|\big)^{p-\delta}
$$
where $X = \sum_{i\in R_t^1}x_i$, etc. The idea is to rewrite the left member of (3.4) as
$$
\big|\E\big[(\phi_1(X)-\phi_1(\E X))\cdot(\phi_2(Y)-\phi_2(\E Y))\cdot(\phi_3(Z)-\phi_3(\E Z))\big] + R\big|,
$$
where $R = \pm\E[\phi_1(X)\phi_2(Y)]\phi_3(\E Z)\pm\dots$ is the remainder we get by algebraically manipulating this expression to match the given left-hand side of (3.4). Writing (3.4) this way sets us up to use the first part of the proof, and indeed the part not involving $R$ agrees with what Bourgain wrote. To see how to handle $R$, I'll just demonstrate what to do with $\E[\phi_1(X)\phi_2(Y)]\phi_3(\E Z)$, since the triangle inequality finishes the bound. By Cauchy-Schwarz,
\begin{align*}
|\E[\phi_1(X)\phi_2(Y)]|\le (\E|\phi_1(X)^2|)^{\frac12}(\E|\phi_2(Y)^2|)^{\frac12}.
\end{align*}
By the bound (3.1), $|\phi_1(X)|\lesssim (1+|X|)^{p_1}$, so by Khinchin's inequality, $\E|\phi_1(X)^2|\lesssim_{p_1}|x|^{2p_1}\le1$. A similar bound applies to $\E|\phi_2(Y)|^2$, of course. To estimate $|\phi_3(\E Z)|$, use (3.1) again to get $(1+|\E Z|)^{p_3}$, and use $p_1+p_2-\delta>0$ to get the final bound $|\phi_3(\E Z)|\lesssim (1+|\sum z_i|)^{p-\delta}$, which is smaller than the right-hand side of (3.4). (Note how the constant doesn't depend on $n$, the number of terms, because we used Khinchin's inequality.)

The last thing to check is $$\big|\E\big[(\phi_1(X)-\phi_1(\E X))\cdot(\phi_2(Y)-\phi_2(\E Y))\cdot(\phi_3(Z)-\phi_3(\E Z))\big]\big|\le \text{RHS (3.4)}.$$
For that, it is helpful to notice, e.g.,
$$
|\phi_1(\E X)-\phi_1(X)| \le C(1+|\sum x_i|)^{p_1-\delta}(1+|\sum (\eta_i-\frac13)x_i|)^p,
$$
which follows immediately from the first line of Bourgain's proof. Ultimately, then, using this inequality, the bound we get is
\begin{align*}
&(1+|\sum x_i| + |\sum y_i| + |\sum z_i|)^{p-\delta}\\
&\quad\times \int(1+|\sum(\eta_i^1-\frac13)x_i|+|\sum(\eta_i^2-\frac13)y_i|+|\sum(\eta_i^3-\frac13)z_i|)^p.
\end{align*}
Use Khinchin's inequality again to show the term on the second line of the above display is $\lesssim_p 1$, finishing the proof.