Ouch - then it's the intersection of two quadrics in $P^4$. I'm not sure what that is: maybe a del Pezzo surface?

The whole of Newman's book was infested by typos, at least the first edition (which I have) was. But there is a second edition, in which I understand most of the typos were corrected (I don't have this).

Dylan, by Siegel's theorem there are only finitely many integer points on an elliptic curve with Weierstrass model over $\mathbb{Z}$. I admit that I haven't worked this problem through to the extent that I am certain it reduces to a problem of this nature, but I suspect it does.

Don't know, but it looks like the obvious bijection induces an isomorphism of fundamental groups.

Many authors cite papers they have never seen, let alone never read :-)

Unless Siegel's method (alas I haven't read the paper) gives $a_d<0$ for $k\equiv2$ (mod 4).

Another reference is Alain Connes's monograph on Noncommutative Geometry, downlodable from alainconnes.org/en/downloads.php .

No, you didn't :-)

Is that a "simple fact"?

It's an exercise (no. 7 in ch. 8).

The Cramer's rule proof does give a little more. If we drop the assumption that $\mathfrak{a}$ is contained in the Jacobson radical it shows that $\mathfrak{a}M=M$ implies that $M$ is annihilated by a ring element congruent to $1$ modulo $\mathfrak{a}$.