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. If you crudely use the triangle inequality on $R$ and apply (3.1) to each of the $6$ terms of $R$, you will end up with, say
\begin{align*}
&6C(1+|\sum x_i|)^{p_1}(1+|\sum y_i|)^{p_2}(1+|\sum z_i|)^{p_3} \\
&\qquad\le 6C(1+|\sum x_i|+|\sum y_i| + |\sum z_i|)^{p}\\
&\qquad\le 6C'(1+|\sum x_i|+|\sum y_i| + |\sum z_i|)^{p-\delta}
\end{align*}
for a different $C'$ that may depend on $p,\delta$. This uses that $|x|,|y|,|z|\le 1$, so all expressions $(1+|\sum x_i|+|\sum y_i|+|\sum z_i|)^q$ are comparable for $q \ge 0$ with constants depending on $q$.

The last thing to check is that after integrating the last displayed expression on p. 235 in $t$ (taking expectation), we get $C(1+|\sum x_i|+|\sum y_i| + |\sum z_i|)^{p-\delta}$. Firstly, the terms
$$
|\sum (\eta_i^1-\frac13)x_i|,\ |\sum (\eta_i^2-\frac13)y_i|,\ |\sum (\eta_i^3-\frac13)z_i|
$$
can each be bounded by $1$ independently of $t$ (again by the assumption $|x|,|y|,|z|\le 1$), and we are left with, say
\begin{align*}
&C(4+|\sum x_i|+|\sum y_i| + |\sum z_i|)^{p-\delta}\\
&\qquad\times\int(|\sum (\eta_i^1-\frac13)x_i| + |\sum (\eta_i^2-\frac13)y_i|+|\sum(\eta_i^3-\frac13)z_i|)^\delta\,dt.
\end{align*}
Now we can simply bound the integral by $3^\delta$ to end up with the right member of (3.4).

The only remark I have about this is that it seems like for the applications, it is perfectly alright to allow the constant $C$ to depend on $p,\delta$.