I fear I won't escape this one :-). I was hoping someone else would reply. The corrections are not complete and I have just taken them off my webpage, not to create more confusion.

If indeed your curve has genus 1, because it has two simple double points as you claim (I have not checked that), then you can transform it into a smooth cubic in the following way. Take a quadratic Cremona transformation based on three points on the curve, two of them being the singular points. The result must be a smooth cubic.

In 3), you should not. This replaces your curve C by another one E such that C->E is a double cover. Now, you can use Hurwitz formular to compute the genus of C if you wish.

As I read it, the first question is equivalent to find one finite extension $L/K$ ramified at most at one place. And I don't see why this is equivalent to the second question.

Unless I misunderstood something: if $X=\mathbb{A}^1_k$, your quotient is just the degree of the extension $k(x_n)\cap k(y_n)$ over $k$.

3. Washington's more recent book has a more on this. 4. is trivial, just take any elliptic curve over F_3 and find them all. Otherwise use sage or magma to list them.

A good place to start to learn about computational number theory (elementary and more advanced) is Henri Cohen's book "A Course in computational algebraic number theory", Spring GTM 138. Algorithm 1.4.3 in there is what is described in this answer.

Yes, there are at least three formulations of the main conjecture for elliptic curves with supersingular reduction at $p$. By Perrin-Riou, by Kato and then by Kobayashi. They are all equivalent. Probably Kobayashi's approach using his $\pm$ Selmer groups linking them to $\pm$ $p$-adic $L$-functions by Pollack is probably the best accessible one.

The algebraic counterparts are characteristic power-series that describe a Selmer group (or class group). They are defined up to a unit power-series. So they are essentially a polynomial, indeed. They are sometimes called "algebraic $p$-adic L-functions". By a main conjecture they should be the same, up to a unit.

Analytic $p$-adic L-function are NOT defined up to a unit. E.g. given an elliptic curve and a prime $p$, say good ordinary, there is one and exactly one $p$-adic L-function. They are good honest functions in one $p$-adic variable $s$. Similar for $p$-adic L-function of Dirichlet character etc.

Finally one should add that it is easy to find the torsion points, if there are any. Most of this is implemented in magma and lots of it in sage.

I agree best with this answer. The descents (as in Robin's answer) tell us that in order to find rational points on an elliptic curve, we better search on one of its torsors. But in the end, we have to do some "brutal search" and that is where the crucial improvements in ratpoints are useful. ... and the only other known method to find rational points is by modularity, say by using Heegner points or variants of them, or (as Pollack and Kurihara do) using supersingular Iwasawa theory. But all of them only work when the analytic rank is 1.

There is a slight speed up: Allow taking negative remainders at each line when they are smaller in absolute value than the positive remainder