This question is a slight generalisation of mathoverflow.net/questions/186807/… . That question resulted in Tyler and me writing a paper on it : arxiv.org/abs/1505.02940

(quickly while on holidays) I got the expansion using sage. doc.sagemath.org/html/en/reference/arithmetic_curves/sage/…. This is an implementation of the description of the formal group in Silverman.

If two elliptic curves have the same formal group law for respective Weierstrass equations, then not only the curves are equal, but even the equations are equal as $F(X,Y) = X + Y -a_1\, XY -a_2\, X^2 Y -a_2\, XY^2 -2a_3\, X^3Y + \cdots + (-2a_1a_3 - 2a_4)X^4Y + \cdots + (-2a_1^3a_3 - 2a_1^2a_4 - 4a_1a_2a_3 - 2a_2a_4 - 2a_3^2 - 3a_6)X^6Y +\cdots$. For any ring of characteristic zero or coprime to $6$.

No, because of twists. In particular over a number field, there are plenty of non-squares $D$ and the quadratic twist $E_D$ is defined over $K$, not isomorphic to $E$ over $K$, but isomorphic over $\mathbb{C}$.

For 2. Yes Shimura curves by the Jacquet-Langlands correspondence. See Chapter 4 in Darmon's book

I don't have access to magma to check what part of the algorithm is random. My guess is: It is rare for a curve to have this property that the algorithm is just at the boundary of succeeding sometimes, but not always. For each curve with this property, it will be difficult to figure out theoretically what the chances are, except of course by just testing 1000 times. Probably one should first study simpler arithmetic functions with random output related to class groups or number fields.

You shouldn't run a random algorithm like this several times with the same arguments; instead you should tweak the optional arguments and finer methods. For instance MordellWeilRankBounds has an argument Effort that you can increase. Definitely you should use analytic methods, like Heegner points, for curves which end up having rank 1.

I guess that one could add a lot to this answer, but the question is maybe not precise enough to know what to include.

Section 2.10 and 2.11 in johncremona.github.io/book/fulltext/chapter2.pdf

Ah. Well that was not clear from your question. You are given the new form and you wish to determine the isogeny class of elliptic curves. That is precisely what was used to create Cremona's tables. The period map determines the set of all periods for all modular symbols (or all closed loops on X_0(N) if you prefer). That gives you one lattice and then you have to check what isogenies are possible.

I think the "elementary methods" refer to the content of the book itself. Nekovář uses the local conditions at places above $p$ to be the Greenberg conditions. I don't know what $p$-adic Hodge theory was/is missing to replace them by $H_f$ conditions in general.

The newform is associated to the full isogeny class of the elliptic curve you start with. The form cannot know which period you wish to obtain, it can only give you the period up to certain rational multiples. Instead, the period is better calculated directly on the elliptic curve, by integrating along the real points of the elliptic curve the absolute value of the Neron differential, that is $dx/(2y+a_1x+a_3)$ in a global minimal model.

If you compare points, you need to fix equations and say these two models of the two varieties intersect in infinitely many points in the ambient projective space. Once these models are fixed there is a choice of the formal group fixed as well and their associated groups can be intersected. Maybe that is what you mean. Certainly the way it is written right now it is very confusing.

If $a_1=1$, then $a_{p^2} - (a_p)^2$ is $\chi(p)\cdot p^{k-1}$. If this is non-zero for a prime $p$, you get $k$.

Part of the definition of an isogeny is that it is a morphism of projective curves. The two maps on $K$-rational points that you are considering are of quite different nature.

As to the question on top: No, being isogenous is an equivalence relation.

It is a local question and over $\mathbb{Q}_2$ you only have finitely many possible extension.

Constructions of abelian extension of real quadratic fields is an active area. See for instance this article by Darmon and Vonk. I fear as stated this question is a bit too broad. What exactly are you looking for?