In the spirit of chapter 5 of Silverman: use that $f:C\to D$ to be an isogeny defined over $\mathbb F_q$ means that $f \circ \phi_C = \phi_D \circ f$, where $\phi_C$ and $\phi_D$ are the Frobenius morphisms on $C$ and $D$ respectively.
Then $$f \circ ( 1_C - \phi_C) = (1_D - \phi_D) \circ f.$$
Take the degree of both sides, and use the fact that $\deg u\circ v = \deg u \cdot \deg v$, and $\deg u\not= 0$ if $u$ is an isogeny. Now, use that $E(\mathbb F_q) = \ker (1 -\phi)$, for any elliptic curve $E$ over $\mathbb F_q$, and that $1-\phi$ is separable.
To answer your second question - I think that you are asking whether the isogeny $f$ over the rationals extends to one (call it $f$ again) over the open set $S$ of $\mathop{\rm Spec} \mathbb Z$ where the two curves have good reduction?
According to lemma 6.2.1 of S's "Advanced Topics in the Arithmetic of Elliptic Curves," a rational map from a smooth scheme to a proper scheme over a dedekind domain only fails to be defined on a set of at worst (at least) codimension 2, "so $f$ extends," and does so uniquely, as implicit in the definitions is 'separated.'
For the extended $f$ to be a group homomorphism one needs that $f$ commute with addition; but that's a Zariski closed condition which holds generically over $S$, so it must hold identically over $S$ (the separated condition). The degree of $f$ doesn't change - use the above to extend the dual isogeny $\check f$, and the relation $f \circ \check f = [m]$, where $m$ is the degree of $f$.
I hope I haven't screwed this up! Even if I haven't, I am sure there are better arguments.
Regarding 37a: When is the product of two consecutive integers, $y$ and $y+1$, equal to the product of three consecutive integers, $x-1$, $x$, and $x+1$.
Is that natural? It's the sort of question one might generalize from $y^2 = x^3$, which is addressed on this site, in which we repeat numbers rather than iterate. What's your notion of naturality?
Best Answer
The local factor of the $L$-function is defined as it is because of the Tate module: it is defined to be $$\det(1-p^{-s}\mathrm{Frob}_p | V_\ell^{I_p})^{-1},$$ where $V_\ell(E)^{I_p}$ is the subspace of the Galois representation $V_\ell(E)$ on which the inertia group $I_p$ acts trivially (so it's the whole of $V_\ell(E)$ if $E$ has good reduction at $p$), viewed as a representation of $G_{\mathbb Q_p}/I_p\cong G_{\mathbb F_p}$.
This definition explains the slightly more ad hoc definition usually given for primes of bad reduction! It has the advantage of being completely uniform.
Since $V_\ell(E)^{I_p}\cong V_\ell(E')^{I_p}$ is isogeny invariant, so is the local factor.