I do not know coherent sheaves over schemes that are not locally noetherian. However, on locally noetherian schemes, the strategy you describe is doubtless right and therefore the answer is yes.

No. You need that the scheme is reduced.

It is certainly true that if $F_x$ is a free $\mathcal{O}_{X,x}$-module of rank $n$, then there exist an open neighborhood $U$ of $x$ such that $F \vert_U$ is a free $\mathcal{O}_U$- module of rank $n$.

But from $\dim_{k(x)} F_x \otimes_{\mathcal{O}_{X,x}} k(x) = 1$ you can deduce that $F_x$ is a cyclic module and not that $F_x$ is free of rank $1$.

However on a reduced scheme the statement is true: exercise II.5.8 of Hartshorne.

The strategy you describe is very common in algebraic geometry and is right in your case.

I do not know coherent sheaves over schemes that are not locally noetherian. However, on locally noetherian schemes, the strategy you describe is doubtless right and therefore the answer is yes.