For any excellent noetherian ring $A$ (which includes any ring finitely generated over a field), by EGA IV$_2$ 7.8.3(v) it follows that for any ideal $I$ of $A$ and the $I$-adic completion $\widehat{A}$ of $A$, the flat map $X = {\rm{Spec}}(\widehat{A}) \rightarrow {\rm{Spec}}(A) = Y$ is regular (i.e., all fibers $X_y$ are regular and remain so after finite extension on $k(y)$). In particular, $\widehat{A}$ is normal if $A$ is normal, so $\widehat{A}$ is a domain in such cases precisely when its spectrum is connected, which is to say ${\rm{Spec}}(A/I)$ is connected.

I believe that Ngo, Ho, and Le Hung have recently carried out a cohomological approach to the function field version of the techniques initiated by Bhargava and Shankar, making good use of the additional geometric structure present in equicharacteristic. Look on the arxiv.

The characterization of when a finite extension (separable or not) admits finitely many intermediate fields is exactly that it is a primitive extension. This is all far more elementary than any categorical formalism.

A more general fact (which can be generalized further) is that if $j:Z \hookrightarrow X$ is a closed immersion between Cohen-Macaulay projective schemes over a field $k$, with respective pure dimensions $d \le n$ then $\omega_{Z/k}=\mathcal{Ext}^{n-d}_X(O_{Z}, \omega_{X/k})$. The key is duality for finite morphisms beyond the finite flat case. The derived category framework (as in Hartshorne's R&D book) illuminates these constructions tremendously (and allows one to study the dualizing sheaf locally, which is ill-suited to the viewpoint of characterization by just a "global" property).

The first question is ill-posed: if $E, E'$ are finite (separable) extensions of $F$ and a connected semisimple $F$-group $G$ splits over $E$ and $E'$ then it generally doesn't split over $E \cap E'$. Unit groups of central simple algebras provide lots of examples over number fields (since the global splitting over a finite extension is controlled by local splitting over a finite set of places). So speaking of "the smallest such extension" doesn't quite make sense (aside from special cases like tori).

@KeerthiMadapusiPera: Without a rational point, "connected" is not "geometrically connected" for schemes over a field. If $K$ is a nontrivial finite extension of $\mathbf{Q}$ then ${\rm{Spec}}(K)$ is a smooth connected $\mathbf{Q}$-scheme with no smooth affine integral model (over $\mathbf{Z}[1/N]$ for sufficiently divisible $N$, say) having all but finitely many fibers connected. Also, without reductivity the special and generic fibers of a smooth affine group with connected fibers over a dvr may have Borels of different dimensions (Bruhat-Tits group schemes!), so Lang's theorem isn't enough.

By the usual tensoring stuff, one can reducing to the case when $(L_2, \nabla_2) = (O_X, {\rm{d}})$ (i.e., the unit object) and $(L_1, \nabla_1) = (M, \nabla_M)$. The relevant calculation is then given early in the book of Mazur and Messing ("Universal extensions and one-dimensional crystalline cohomology").

@user75148: It does prove what you asked. I'm not aware of any specific reference with this result. If you have a friend or colleague who knows a bit more commutative algebra, perhaps if they look at this answer then they can try to explain it to you in terms that are more accessible (I don't know your background). As a student, I found that many puzzling facts about Dedekind domains became far more transparent once I learned some more commutative algebra, to enable me to attack questions about ideals in the more robust framework of abstract projective and invertible modules.

@Aleksa: If $\rho:G \rightarrow {\rm{GL}}(V)$ is a finite-dimensional irreducible representation over an algebraically closed field then the unipotent radical acts trivially, so it factors through the maximal reductive quotient. So really the case with reductive identity component is the essential one (for smooth affine groups over an alg. closed field).

Is there some motivation for this question (i.e., an application in mind)?

Just to clarify, Jim Humphreys' comment is referring not to the general coset space construction as in this answer, but rather to the finer assertion of the affineness property when $H$ is normal in $G$. (His reference is to Chapter II of Borel's book.)

@S.Carnahan: Very good. Every time I try to dig up that reference I have a devil of a time tracking it down again inside DG because it is inside a section called "Commutative groups" which I skim past, thinking it cannot be in there...though the subsection name in the table of contents gives it away if one looks closely enough.

This characterization over fields with positive characteristic is due to Nagata from long ago (Olsson-Vistoli extend it to work over rings/schemes); this is proved in somewhat "elementary" terms somewhere in the huge book of Demazure-Gabriel, but I am unable to relocated the exact location in there at the moment.

And likewise for every (smooth proper) curve over a finite field by class field theory (though it seems that the OP is only asking about varieties over an algebraically closed field).