I'm not sure I quite understand what you intend to do with EdgeAdd
thus I will only answer the "I want to modify the edge 1-4 of the CompleteGraph[5]
" question.
Here I modify the edge 1 <-> 4
and 1 <-> 2
for generalisation purposes:
g = CompleteGraph[5];
weightsVal = {a, b}; (* in the same order as {1 <-> 4, 1 <-> 2}*)
weights = ReplacePart[Array[1 &, Length@EdgeList@g],
Rule @@@ Thread@{Flatten[Position[EdgeList@g, #] & /@ {1 <-> 4, 1 <-> 2}], weightsVal}]
Graph[EdgeList@g, AbsoluteOptions[g, VertexCoordinates],
EdgeWeight -> weights,
EdgeLabels -> Rule @@@ Thread@{EdgeList@g, weights},
VertexLabels -> "Name", ImagePadding -> 10]
% // WeightedAdjacencyMatrix // MatrixForm

$\left(
\begin{array}{ccccc}
0 & b & 1 & a & 1 \\
b & 0 & 1 & 1 & 1 \\
1 & 1 & 0 & 1 & 1 \\
a & 1 & 1 & 0 & 1 \\
1 & 1 & 1 & 1 & 0 \\
\end{array}
\right)$