Now it is a duplicate of this question. There this field is shown as an example of a degree 6 extension of $\mathbb{Q}_2$ with $e=2$ but only a unique quadratic subfield, which is unramified over $\mathbb{Q}_2$.

In the online documentation of sage you will find functions to work with Tate curves, including the function lift that is your $\phi^{-1}$. I implemented the map to components a long time in this script as component_of_a_point(P,v)

Look up exceptional sequences and exceptional units for this question in a fixed number field.

Just do examples. $y^2+xy=x^3+px+p$ needs no blowups and is of type I${}_1$. What happens to $(0,\sqrt{p})$ ?

"any" is ambiguous in English. Here it means "in at least one" fibre, not in all fibres. Yes, to lie in a component is so say that the section intersects that component.

I believe the above is what Will meant. I hope that clears up your main problem.

This interpretation is correct. The $m$-torsion of $E(\bar K)$ is generated by the classes of $\zeta_m$ and $\sqrt[m]{q}$ in $\bar K^{\times}/q^{\mathbb{Z}}$ where $\zeta_m$ is a primitive $m$-th root of unity. Extend the valuation $v$ of $K$ to $\bar{K}$. Then if $v(T)$ for a $T$ in $\bar K^{\times}$ representing an $m$-torsion point is not an integer, then it is not in $E(K)$. Hence it is not visible in the Néron model of $E/K$ as lying on one of the components. You would need to go to a larger field and a new Néron model over that field.

Your notation $\mathbb{Z}_p$ is probably not the $p$-adic integers, but $\mathbb{Z}/p\mathbb{Z}$.

Yes, there is the method .saturation. That works if the algorithm has found points that generate a subgroup of finite index. However over a large degree number field, it can very well happen that the heights of the generators are so large that the search won't find them.

You confuse local and global.

Sure, but if $p$ divides $v(q)$, we could have that $q$ is a $p$-th power in $K_v$ in which case $K_v=L_w$ is a trivial extension. The important part is that the extension is non-trivial of degree $p$.

I believe this question can be closed once you understand the answer to your first question.

The sage question: Sage has Denis Simon's gp script for calculating the rank of elliptic curve over number fields, but the search for points is often more efficient using magma. It turns out this is often the hardest part of evaluating your right hand side. Once you have the points it is not hard to decompose it into irreducible factors using characters. But none is systematically implemented.

Good question. I don't think there is a large amount of examples like this ever been computed or written down. I might have some, I will have to search for them. There is a folklore conjecture which states that ranks should be 100% of the time minimal (0 or 1 imposed by parity conditions), which will tell you that indeed it is rare to find examples of your kind. But over $\mathbb{Q}$ we also find rank $2$ curves.

The answers to this question discusses how to compute it for a given $n$.

$j(E)\in\mathbb{Z}$ is a very strong assumption.

You are probably right and this is a typo in this note for CRAS. It doesn't affect the validity of the argument that follows. Or maybe the author has a different convention for the sign of a discriminant.

The answer to your "Is it true?" question is "No". Take any Weierstrass equation $y^2=x^3+Ax+B$ with integers $A$ and $B$. Now $Y^2 = X^3 +A\ell^4 x + B\ell^6$ is also a model whose reduction is clearly additive at $\ell$, yet the reduction type of the initial equation (which could be minimal at $\ell$) could be anything.