There have been substantial generalizations of this result (by Shou-wu Zhang et al.) which put it into a broader framework, making more effective use of automorphic methods to reduce the dependence on quirky facts about GL(2) and modular curves, etc. So our understanding of what underlies the theorem is much improved, though not sure any of it should be called a dramatic "simplification": still very hard stuff.

Kestutis is correct: there are cohomological obstructions to lifting maps without the affineness hypothesis on $Y$. On the other hand, for affine $Y$ the lifting problem can be checked Zariski-locally since the relevant obstruction to globalizing lives in a higher quasi-coherent cohomology group on $Y$ that vanishes since $Y$ is affine.

Since $R$ is local, it is an inverse limit of finite local rings, all with the same residue field, and its maximal ideal is the inverse limit of those maximal ideals, so $m$ is closed in $R$ with finite index, hence automatically open. The same analysis shows that $R$ is $m$-adically separated since each artinian quotient from the inverse-limit description is max-adically separated. Of course higher powers of $m$ need not be closed (let alone open). Since some power of $m$ vanishes in each artin local quotient of $R$, $m$-adic completeness holds too.

@Niels: I assumed that anyone posting a question such as the above had already tried looking at the relevant parts of SGA1 and for whatever reason didn't find that sufficient on its own, so I decided to offer an alternative viewpoint which I found instructive to think through when learning these things. Of course, the OP might find my answer to not be helpful; maybe we'll find out some day. :)

So this approach is more elementary than my answer since it doesn't use Serre's theorem, and only requires some basic properties of flatness and results known in the pre-homological era (though of course the viewpoint of formal smoothness only came along later).

To provide relevant references in Matsumura's "Commutative Ring Theory" (EGA 0$_{\rm{IV}}$, 19.5--19.6 is similar, but with heavier style), the "i.e." is Thm 26.9 and by flatness of $A\rightarrow A\otimes_k K$ and regularity of $A$ it suffices (by dimension formula for flat local maps; see Thm 23.7(ii)) to prove regularity of local rings of the noetherian fiber algebras. A local noetherian ring $(R,m)$ formally smooth for the discrete topology over a field $F$ is formally $F$-smooth for its $m$-adic topology, so conclude via Cohen's work on coefficient fields: Thm 28.3 and subsequent Lemma 1.

@Tomasz Lenarcik: If $R$ is a regular local $k$-algebra of dimension $d$ (e.g., localize $A$ at a prime), $\{t_1,\dots,t_d\}$ generates its maximal ideal $m$, and $K$ is finite separable over $k$, then $K\otimes_k (R/m)$ is a finite direct product of fields: residue fields at maximal ideals of the semi-local $K\otimes_k R$. Thus, $\{t_1,\dots, t_d\}$ generates each local ring of $K\otimes_k R$ at a maximal ideal (!). Those local rings have dimension $d$ (since $K\otimes_k R$ is finite flat over $R$), so they're regular. In fancier terms, an etale algebra over a regular ring is regular.

By uniqueness of real closures up to isomorphism, every order structure on a number field (if any exist) arises from an injection into $\mathbf{R}$ as a subfield. That addresses the question at the end, but I'm not sure what the first paragraph (linear orders and Sturm's algorithm, etc.) has to do with this.

Since you wrote "i.e., permute the rows of $x$" I thought you meant that $\pi$ is the operation of permuting rows, so $V$ is contained in the open locus of matrices with non-vanishing upper-left entry (that much gives $L \times U$) which moreover retain that property after swapping of the rows. That $V$ has empty intersection with $L$ and $U$. Please describe an explicit $V$ that meets $L$ and $U$.

@Andre: Going beyond being a "bad definition", is it even a definition if one hasn't already defined what "locally looks like" is supposed to mean (e.g., already have introduced a suitable 2-category in which these objects are meant to live, providing an a-priori notion of 1-morphism than makes the question posed somewhat moot)? Perhaps this is also implicit in your phrase "bad definition"...

Have you tried to compute this for $n=2$? The open locus $L \times U \subset {\rm{GL}}_2$ is given by non-vanishing of the upper-left entry if I'm not mistaken, but if you consider points of either $L \times \{1\}$ or $\{1\} \times U$ you see that nothing in either of these retains that non-vanishing property after swapping the two rows. Thus, $L \cap V$ and $U \cap V$ are empty when $n=2$.

Let $G$ act on itself by conjugation, so you ask if $Q:=(G.g)(F)/G(F).g$ is finite. Since $G\rightarrow G.g$ is a right $Z_G(g)$-torsor (for etale topology if $Z_G(g)$ is smooth, fppf in general), by Cor. 1 to Prop. 36 in section 5.4 of Ch. I of Serre's "Galois cohomology" and its fppf variant we see that $Q=\ker({\rm{H}}^1(F,Z_G(g))\rightarrow {\rm{H}}^1(F,G))$. By 4.3 in Ch. III of that book, ${\rm{H}}^1(F,H)$ is finite if $F$ has characteristic 0; if char$(F)>0$ then finiteness holds if $H$ is connected reductive (rests on Bruhat-Tits) but fails in the smooth connected commutative case.