Should $E$ and $B$ change with Gravity? Lets examine a typical GR metric:
$$ds^2=g_{00}dt^2-g_{11}dx^2-g_{22}dy^2-g_{33}dz^2$$
The "d" going with ds has its correct meaning when the path is
specified with respect to a one dimensional manifold (remembering that ds
is the proper time interval which will depend on path).
The physical distance (spatial) between two points along the x-axis
between the points A and B is given by:
$\int\sqrt{g_{11}}dx$ from A to B and not by $\int dx$ in curved space.
Infinitesimal separation between points on the x axis are given by
$g_{11}dx$ and not by $dx$.
Now in Maxwell's equations in the covariant form we have quantities
like delta-x,delta-y etc which are meaningful only in the
Euclidean (rather in the flat space-time Lorentzian) context.But
Maxwell's equations in the covariant form refer to curved space
time (with respect to strongly curved spacetime also). Are these quantities
($\partial x$,$\partial y $ etc.) expected to retain their physical significance
in curved space-time except that they remind us of an Euclidean background?
Better we could write (locally):
$$ds^2=dT^2-dX^2-dY^2-dZ^2$$
Where,
$$\begin{align}dT&=\sqrt{g_{00}}dt \\ dX &= \sqrt{g_{11}}dx \\ dY &= \sqrt{g_{22}}dy \\ dZ &= \sqrt{g_{33}}dz\end{align}$$
(The "d" going with T,X,Yand Z is as justified as the d going with s.)
Locally we have,
$$ds^2=dT^2-dX^2-dY^2-dZ^2$$
Therefore locally we have the same form of  Maxwell's equations--
Maxwell's equations in the traditional form!
Though the form of Maxwell’s equations (traditional form  being
referred to here) remains unchanged locally, the values of the
individual variables may change, preserving the traditional form of
Maxwell’s equations in the local inertial frames.
We may consider a pair of  local labels $x$ and $x+dx$. The
physical distance between them along the x-axis  is $g_{11}dx$. If the
metric changes, say due to the advance of a heavy mass or a high
density mass distribution, the physical intervals $dX$,$dY$ etc will
change. To preserve the form of the equation the values of $E$,$B$,$j$ etc
should also change.
So gravity can change the magnitudes of $E$,$B$ etc. (and of course their directions). If one thinks in the cosmological direction the curvature of space-time was very strong in
the remote  past and gradually it weakened casting a heavy influence
on the values of the electric and the magnetic fields.
Query: How are they taking care of this in the LHC experiments in tracing the past?
[Incidentally the quantities x,y,z etc are simply labels in the curved spacetime context. dx should correspond to some "Euclidean memory"]
 A: A lot of the things you wrote are actually somewhat close to some modern notations and methods used in GR.  However, first there are a few things that need to be mentioned.  First of all, your metric is far from typical, The majority of Einsteinian metrics are not diagonal as you first mentioned.  There really should be 10 terms in that sum, not four...  Secondly, I would like to alert you to quantities in tensor analysis known as "tensor densities".  These quantities come "weighted" with modified coefficients of the metric.  There are scalar densities, vector densities etc... and slightly resemble your redefined terms in the metric, though their construction is a bit more rigorous.  Finally, Maxwell's equations can certainly be written in a form that jives perfectly well with Einsteinian manifolds.  Two of them are
$$
\nabla_{\mu}F^{\mu \nu}=J^{\nu}
$$
They just look the same but instead we replace the partial derivative with the covariant derivative.  These are suited for use in an arbitrary Einsteinian manifold.
I hope this helps,
A: 
Query: How are they taking care of this in the LHC experiments in tracing the past?

LHC experiments are searching for new particles and interactions at an energy scale which describes a time slice of the evolution of matter after the Big Bang. You may say that it is searching for knowledge of how matter evolved quantitatively in the lab. Much higher energies are displayed by cosmic rays and studied with long arrays of detectors in other experiments. Experiments are studying  and searching for constants of motion : masses, spins, decay chains ...
E and B are not constants of motion and it is not surprising that they would not be so when GR is included. The values of the constants of motion searched and the correlations studied are not affected by any E and B dependence . That is why we study constants of motion, instead of recording the changes in the electric field of the proton or the magnetic field of the neutron  or .... Invariants is the game.
