Two questions on isomorphic elliptic curves

Question 1: Putting both curves in say, Legendre Normal Form (or else appealing the lefschetz principle) shows that if the two curves are isomorphic over $\mathbf{C}$ then they are isomorphic over $\overline{\mathbf{Q}}$. Now we could say that for instance $E_2$ is an element of $H^1(G_{\overline{Q}}, Isom(E_1))$ where we let $Isom(E_1)$ be the group of isomorphisms of $E_1$ as a curve over $\mathbf{Q}$ (as in Silverman, to distinguish from $Aut(E_1)$, the automorphisms of $E_1$ as an Elliptic Curve over $\mathbf{Q}$, that is, automorphisms fixing the identity point). However, $E_2$ is also a principal homogeneous space for a unique curve over $\mathbf{Q}$ with a rational point, which of course has to be $E_2$, so the cocycle $E_2$ represents could be taken to have values in $Aut(E_1)$. Now $Aut(E_1)$ is well known to be of order 6,4 or 2 depending on whether the $j$-invariant of $E_1$ is 0, 1728 or anything else, respectively. Moreover the order of the cocycle representing $E_2$ (which we now see must divide 2, 4 or 6) must be the order of the minimal field extension $K$ over which $E_1$ is isomorphic to $E_2$. So $K$ must be degree 2,3,4 or 6 unless I've made an error somewhere.

Question 2: If you restrict your focus to just elliptic curves, yes your idea is right. If it's a quadratic extension, you have exactly 1 non-isomorphic companion. If you have a higher degree number field, you have nothing but composites of the quadratic case unless your elliptic curve has j invariant 0 or 1728.

Notice I am very explicitly using your choice of the word elliptic curve for both of these answers.

The answer is a bit more complicated if $j=0,1728$ because the corresponding elliptic curves have a bigger automorphism group, so I'll leave those out and let you (or others) deal with this case. If $j \ne 0,1728$, then the automorphism group of $E$ is of order $2$ and all other elliptic curves isomorphic to $E$ over an extension are quadratic twists. It seems that you know what happens in this case. This is well-discussed in Silverman's book.

Hopefully I'll have some time later to ellaborate, but for now - here is a great reference I wish somebody had shown me when I started out, for how to attack questions of this type: http://books.google.com/books?id=l0DgAIx_djoC&printsec=frontcover&dq=waterhouse+affine&source=bl&ots=nup7qU4Aln&sig=VXgsPcQNgAkCbNsH8KG-VAakjio&hl=en

look at chapter 17 (and 18).