I'm reading the proof(s) of Mordell-Weil theorem using various texts. This post is to make sure what I'm reading and understanding is correct..

Rational points on Elliptic Curves by Silverman Tate gives a proof of Mordell 's theorem when elliptic curves are over rationals, $\mathbb{Q}$and have at least one rational $2$-torsion point.

Then I noticed that Lawrence Washington's book on Elliptic curves also discuss this result this, first when all roots of cubic $f(x)$ are in $\mathbb{Q}$ and then just one of them is in $\mathbb{Q}$. And the same goes for Cassels' and Knapp's book as well.

Then Lawrence Washington gives a remark that same can be said (i.e., Mordell-Weil holds) when we consider elliptic curves ($y^2 = f(x)$) over any number field $K$ and $f(x)$ has all roots in $K$ as we can prove Weak Mordell Weil in this case with some modification of the usual $\mathbb{Q}$ argument. But then it says nothing about the case when we just assume that $f(x)$ has at least one zero in $K$.

Now my questions,

(1) I'm assuming that we can give a proof of Mordell-Weil when elliptic curves are over K and $f(x)$ has (just) at least one point in $K$. And I think Silverman Tate's proof in case of $\mathbb{Q}$ can be generalized, mainly Weak Mordell-Weil's (Lemma $3.4$ in the book)

(2) What happens when elliptic curves ($y^2=f(x)) E(\mathbb{Q})$ and $E(K)$ have no $2$-torsion point or $f(x)$ has no root in $\mathbb{Q}$(respectively $K$) ? Is it the situation that is discussed in the J.S.Milne's Elliptic Curves book or J. Silverman's The Arithmetic of Elliptic Curves?

I would appreciate if someone could help answer some or all of the questions above or maybe just refer to an appropriate source to find these proofs/ discussions.

Thank you!