On the equality case of the Hölder and Minkowski inequalites
On leo's request I'm posting my comment as an answer.
Your treatment of the equality cases of Hölder's and Minkowski's inequalities are perfectly fine and clean. There's a small typo when you write that $\int|fg| = \|f\|_p\|g\|_q$ if and only if $|f|^p$ is a constant times of $|g|^q$ almost everywhere (you write the $p$-norm of $f$ and the $q$-norm of $g$ instead).
The case where either one $\|f\|_p$ or $\|g\|_q$ (or both) are infinite isn't part of this exercise and simply wrong. You can trisect $E = F \cup G \cup H$ into disjoint measurable sets of positive measure, take $f$ not $p$-integrable on $F$ and zero on $G$, take $g$ not $q$-integrable on $G$ and zero on $F$ and choose $fg$ non-integrable on $H$. Then certainly no power of $|f|$ is a constant multiple of a power of $|g|$ and vice versa, even though equality holds in the Hölder inequality.
A very nice “blackboard summary” of the equality case (for finite sequences) is given in Steele's excellent book The Cauchy–Schwarz Master Class. Let $a = (a_1,\ldots,a_n) \geq 0$ and $b = (b_1, \ldots, b_n) \geq 0$ and let $\hat{a}_i = \dfrac{a_i}{\|a\|_p}$ and $\hat{b}_i = \dfrac{b_i}{\|b\|_q}$. Then your argument is subsumed by the diagram (with an unfortunate typo in the upper right corner—no $p$th and $q$th roots there):
Mimicking this for functions, let us write $\hat{f} = \dfrac{|f|}{\|f\|_p}$ and $\hat{g} = \dfrac{|g|}{\|g\|_q}$ (assuming of course $\|f\|_p \neq 0 \neq \|g\|_q$), so $\int \hat{f}\vphantom{f}^p = 1$ and $\int \hat{g}^q =1$ and thus your argument becomes $$ \begin{array}{ccc} \int |fg| = \left(\int|f|^p\right)^{1/p} \left(\int|g|^q\right)^{1/q} & & |f|^p = |g|^q \frac{\|f\|_{p}^p}{\|g\|_{q}^q} \text{ a.e.}\\ \Updownarrow\vphantom{\int_{a}^b} & & \Updownarrow \\ \int \hat{f}\,\hat{g} = 1 & & \hat{f}\vphantom{f}^p = \hat{g}^q \text{ a.e.} \\ \Updownarrow\vphantom{\int_{a}^b} & & \Updownarrow \\ \int \hat{f}\,\hat{g} = \frac{1}{p} \int \hat{f}\vphantom{f}^p + \frac{1}{q} \int \hat{g}^q & \qquad \iff \qquad & \hat{f}\,\hat{g} = \frac{1}{p} \hat{f}\vphantom{f}^p + \frac{1}{q} \hat{g}^q \text{ a.e.} \end{array} $$
I suggest that you draw a similar diagram for the equality case of Minkowski's inequality.
I'll add some details on the Minkowski inequality (this question is the canonical Math.SE reference for the equality cases, but almost all of it concerns Hölder's inequality).
The standard proof of the Minkowski inequality begins with $$ \begin{align*} \int |f+g|^p &\le \int |f||f+g|^{p-1} + \int |g||f+g|^{p-1} \\ &\le \|f\|_p \| |f+g|^{p-1}\|_q + \|g\|_p \| |f+g|^{p-1}\|_q \end{align*} $$ where $q$ is the conjugate exponent to $p$. This simplifies to $\|f+g\|_p^p \le (\|f\|_p+\|g\|_p) \|f+g\|_p^{p-1} $ as wanted. So, if equality holds, it also holds in the two instances of Hölder's inequality above. Hence $|g|^p$ and $|f|^p$ are both constant multiples of $(|f+g|^{p-1})^q$, which makes them collinear vectors in $L^1$.
Additionally, the equality case requires $|f+g| = |f|+|g|$, which means the signs (or arguments, in the complex case) of $f$ and $g$ must agree a.e. where the functions are not zero. Conclusion: $f$ and $g$ are collinear vectors in $L^p$.