Question 1: In the answers to a previous question of yours, I think it was explained that one way to view a connection is as infinitesimal descent data. It might be more useful, however, to look at some specific examples: when $G=GL_n$, then a $G$-bundle with flat connection is just a vector bundle of rank $n$ with a flat connection; when $G=SL_n$, it's a vector bundle $V$ of rank $n$ equipped with a flat connection $\nabla$ such that $\det(V)$ admits a $\nabla$-parallel generating section; when $G=SO_n$, it's just a $SL_n$-bundle with flat connection $\nabla$ equipped also with a $\nabla$-parallel non-degenerate quadratic form....

Question 2: In any of the above explicit settings, giving a reduction to a Borel $B$ simply amounts to giving an appropriate full flag of sub-bundles for $V$ (for example, if $G=SO_n$, you will require some isotropicity condition for the flag). A connection will preserve the reduction $\mathfrak{F}_B$ if it preserves the flag it corresponds to (in the sense that every derivation preserves the flag). If $\nabla$ and $\nabla'$ are two flat connections (of $G$-bundles), then their difference is an element of $End(V)\otimes\Omega^1$. Of course, their difference will kill any structures on $V$ that are parallel with respect to both $\nabla$ and $\nabla'$ : so, if $G=SL_n$, then it will be of trace $0$; if $G=SO_n$, then it will be of trace $0$ and preserve the quadratic form. This sub-space, which is a twist of $\mathfrak{g}=Lie(G)$ (viewed as a representation of $G$) by the $G$-torsor $\mathfrak{F}$, is what is being referred to as $\mathfrak{g}_{\mathfrak{F}_G}$.

So we find that $\nabla-\nabla'$ is an element of $\mathfrak{g}_{\mathfrak{F}_G}\otimes\Omega^1$.

EDIT: I was trying to give an idea of what preserving $\mathfrak{F}_B$ meant, but the notion of $\nabla$ preserving $\mathfrak{F}_B$ is simple in general: we have an identification of $G$-torsors:
$$\mathfrak{F}_G=G\times^B\mathfrak{F}_B.$$
We now simply require the connection $\nabla$ to arise from a connection on $\mathfrak{F}_B$ via this identification.

Question 3: Here, I think they're only asking for a local trivialization of $\mathfrak{F}_B$. A trivialization of $\mathfrak{F}_B$ gives an isomorphism of $B$-torsors from $B\times X$ to $\mathfrak{F}_B$. The former has a tautological connection over $X$, since it's a constant $X$-scheme. EDIT: The local trivialization gives a local tautological connection!

Question 4: No idea.

EDIT: In general, if you have a $G$-torsor $P$ over $X$, and a representation $V$ of $G$, you can define the twist $V_P$ as a vector bundle over $X$: this is the contraction product $$V\times^G P=(V\times P)/G,$$
where $G$ is acting diagonally. If you look at 1.2.4 in Frenkel's book ('Langlands correspondence...'), then you'll find that giving a connection on $P$ is equivalent to giving compatible flat connections on the twists $V_P$ of all algebraic $G$-representations $V$. Essentially, if you have infinitesimal descent data for $P$, then you clearly have corresponding data for the twists $V_P$; the converse is trickier and uses some Tannakian theory. When $G$ has a faithful representation $V$ (like in our examples), it is enough to do this for the one twist $V_P$ (again, by the Tannakian machine).

Quantization of Hitchin's integrable system and Hecke eigensheaves– S. Carnahan♦ May 24 '11 at 5:59