Unipotent vector bundles Let $X$ be a scheme which is smooth over a noetherian base scheme $S$.
Let $(\mathcal F,\nabla)$ be a flat vector bundle, i.e. $\mathcal F$ is a vector bundle of finite rank on $X$ and $\nabla$ is an integrable connection relative $S$.
Is there a notion in the literature of "unipotent" flat vector bundle and morphisms between them? 
And does this give an abelian category with objects isomorphism classes of such $(\mathcal F, \nabla)$? I haven't seen a clean formal approach to this in the literature.
I don't think it is abelian in the above wide context, but probably for $X$ being a smooth variety over a field or an abelian variety over a field.
 A: As Keerthi Madapusi Pera points out in his comments, it is certainly reasonable to define  a unipotent flat vector bundle as a flat vector bundle that is a successive extension of the trivial one $(\mathcal O_X,d)$. Over a general basis $S$ I don't know, but over a field there are plenty of references.
For instance let $k$ be a field of characteristic zero and $X$ a smooth geometrically connected scheme over $k$. Then according to
Deligne, P.
Le groupe fondamental de la droite projective moins trois points.
Zbl 0742.14022
$\S$ 10.26 the category of unipotent flat vector bundles over $X$ is even Tannakian (a fortiori abelian), giving rise to the "De Rham fundamental group" $\pi(X,x)_{DR}$.
If you want to work in positive characteristic, or with singularities, flat vector bundles are certainly not the right objects any longer (at least if you want to have some link with the fundamental group), one has to replace them with stratifications, see
Saavedra Rivano, Neantro
Catégories Tannakiennes.
Lecture Notes in Mathematics, Vol. 265.
Zbl 0241.14008
VI 1.2
or
dos Santos, João Pedro Pinto
Fundamental group schemes for stratified sheaves
J. Algebra 317 (2007), no. 2, 691–713.
Zbl 1130.14032
One could also mention that over a complete scheme over a field, you don't need to consider connexions at all. For instance Nori in chapter IV of his PhD
Nori, Madhav V.
The fundamental group-scheme.
Proc. Indian Acad. Sci. Math. Sci. 91 (1982), no. 2, 73–122.
Zbl 0586.14006
considers a scheme of finite type $X$ over a field $k$ such that $H^0(X,\mathcal O_X)=k$, and proceeds to show that the category of unipotent vector bundles is indeed Tannakian (the explanation is that one can deduce from this that unipotent vector bundles are in fact endowed with connections). Lemma 2 there may be of interest for you, because he shows exactly the abelianness in this similar context.
A: Your last question "Furthermore, another question is if any extension of a flat vector bundle by unipotents is again unipotent." can be partially answered by the following two papers:
"A remark on "Connections and Higgs fields on a principal bundle""
http://arxiv.org/abs/1102.4216
"Unipotent Schottky bundles on Riemann surfaces and complex tori"
http://arxiv.org/abs/1102.3006
