Differentials of the second kind
Gross and Rohrlich in the paper On the periods of abelian integrals and a formula of Chowla and Selberg state the claim below without citation (pg. 198), giving an explicit determination of the cohomology classes of the Fermat curve $X_d := \{x^d + y^d - z^d = 0\} \subset \mathbb{P}^2$ using differentials of the second kind.
Working on the affine chart $\{z=1\} \subset \mathbb{P}^2$ they define the following meramorphic 1-forms on $X_d$: $$ \eta_{r,s,t} = x^{r-1}y^{s-d} \mathrm{d} x. $$ These forms can be integrated over homology classes, giving the cohomology classes that we need.
Claim: The middle cohomology of $X_d$ is generated by $\eta_{r,s,t}$ where $0 < r,s,t < d$ and $r+s+t \equiv 0 \,(\text{mod } d)$.
Example: Let $d=3$. Then, $X_3$ is elliptic and we have $\mathrm{H}^1(X_3,\mathbb{C}) = \langle \frac{\mathrm{d}x}{y^2} , \frac{x\mathrm{d}x}{y}\rangle$.
Main problem: How does one prove the claim above?
Perhaps using Griffiths residues
We could try to prove this statement using the Griffiths' residue map: $$ \operatorname{res}: \bigoplus_{\ell \ge 1} \mathrm{H}^0(\mathbb{P}^2,\Omega_{\mathbb{P}^2}(\ell X_d)) \to \mathrm{H}^1(X_d,\mathbb{C}), $$ which implies that the residues of the following meramorphic forms generate the image: $$ \omega_{r,s,t} = \frac{x^{r-1}y^{s-1}z^{t-1}}{(x^d+y^d-z^d)^\ell }\cdot \Omega, $$ where $0 < r,s,t < d$, $r+s+t = \ell d$ and $\Omega = x \mathrm{d}y\mathrm{d}z - y \mathrm{d}x\mathrm{d}z + z\mathrm{d}x\mathrm{d}y$.
Therefore it remains to integrate the forms $\omega_{r,s,t}$ over a suitable tubular neighbourhood of $X_d$. Representation theoretic arguments imply that $\operatorname{res} \omega_{r,s,t} = c_{r,s,t} \eta_{r,s,t}$ for a scalar $c_{r,s,t} \in \mathbb{C}^*$, provided Gross and Rohrlich are correct.
However, an explicit computation reveals that the residue of $\omega_{r,s,t}$ is in fact a scalar multiple of $x^{r-1}y^{s-\ell d} \mathrm{d}x$ instead of $x^{r-1}y^{s-d} \mathrm{d}x$. Therefore we can answer Question 1 if we can answer the following.
Equivalent problem: How can we lower the pole order of $x^{r-1}y^{s-\ell d} \mathrm{d}x$ modulo cohomological equivalence to get $x^{r-1}y^{s-d}\mathrm{d}x$?
