Effect of single-gene overexpression in the cell's response
Which are the factors that modify the overall gene differential
  expression by introducing a vector for single-gene overexpression


This is a very relevant question and the field of experimental biology needs to revisit the experimental strategies.

One of the explanations that you gave is correct- that the overexpressed protein might override the cellular processes. 

Another problem that this practice gives rise to is incorrect inference. We are mostly interested in knowing what a protein does in a cell at general physiological conditions. If the stoichiometry is altered wrong results will definitely pop up.
For example in the case of a repressor/activator with multiple targets: At its general physiological concentration it may not really activate gene-X because of low affinity but in high concentrations it just may, and we end up concluding that gene-x is a target of repressor-1 by an OVEREXPRESSION experiment. 

A synthetic biological approach is very good in studying small signaling/transcriptional modules. The entire module can be cloned and expressed in a cell under an inducible/constitutive promoter. This can be implemented in a heterologous cell also. A mathematical model generally helps in making certain predictions. 

Other strategies which can be used instead of overexpression:


using sensor constructs harboring a functional or non-functional
target sites instead of overexpressing the upstream protein
downregulation experiments