From an algebraic geometry point of view: the surfaces I-III are rational surfaces. I.e., they contain an open that is isomorphic with an open in $\mathbb{P}^2$. Several rational surfaces come with a name, but most of them are smooth projective (del Pezzo surfaces, quadric surfaces), whereas your surfaces are singular. Examples of rational surfaces "come with a name" are cones over rational curves, surfaces swept out by a morphism between two curves. You can check whether your surfaces are of one of these types.

I read large part of Eisenbud's book, but I could not find it in there.

The weights for $H^{2n−i}(X_{smooth})$ lie in $[2n−i,4n−2i]$, therefore the weights for $H^{2n−i}(X_{smooth})^∗(n)$ lie in $[2i−2n,i]$. If $i>2 dim X_{sing}+1$ and $i>n$ this yields a non-trivial bound on the weights for $H^i_c(X)$ and $H^i(X)$.

- I attended a talk by Steve Donnelley in 2005, where he explained how one can get d_p<p. He applied a two-descent to a subclass of the examples of large p-Selmer groups that Ed Schaefer and I constructed. At the time he did not work out a formula for $d_p$, but I figure that you get something like $d_p \sim 1/12 p$. Unfortunately, I never saw a written account of this. There is a further argument that shows that the examples of Ed and my are examples of big sha. But I did not write that up yet. - If you fix besides $p$ also $E$ (as you and Sharif did) then $d_p\geq p$ holds.

However, for every prime number $p$ and every integer $k$ there is an abelian variety $A/\mathbb{Q}$ such that the $p$-torsion in the Tate-Shafarevich group has $\mathbb{F}_p$-dimension at least $k$. In this construction the dimension of $A$ grows with $p$.

There are several further references for explicit calculation of Picard-Fuchs. E.g., there are papers by Peters, Beukers-Stienstra or Verrill. More recently explicit calculation of the (p-adic) PF equation is used by people in point count algorithm. There is a paper by Alan Lauder on average ranks of elliptic curves over function fields. This paper is based on algorithm that calculates solutions to the p-adic PF equation.