It looks like you have gotten very good answers to most of your questions from those above.
About your question, a) Is that possible that the excitation system allows the existence of unbalanced active and reactive power generation between the phases?
As far as I know, the answer is NO. The excitation system acts to increase the magnetic field of the rotor, which increases the voltage (very simple explanation), the balance is mostly affected buy the connected loads. Some newer design generators, with what is called "square laminations" by one manufacturer, has an inherent votlage imbalance, and in networked systems canhave some strange readings at times, but not usually a problem, except if you're unfamiliar with it it may cause some head scratching.
As well explained above, increase excitation from the AVR to the pilot excitor (I'm assuming this is what you have due to size and age you have described)and the terminal voltage will rise if operating in a standalone condition. If you are networked, or paralleled, the network pretty much defines the system voltage, unless your generator has enough capacity to change the system. So your AVR can't work very well for controlling voltage. Most power factor controllers for your size unit work by becoming active when told the unit is in parallel, then biasing the voltage adjustment reference setpoint. If the unit is under-excited and consuming VAR's, the controller raises the reference setpoint, causing the AVR to increase excitation, in turn causing the generator to consume less VAR's, or produce VAR's into the system. If the unit is over-excited and producing excess VAR's, then the controller reduces the setpoint, the AVR lowers the excitation level, and so on.
Along with this is the dynamic adjustments, the AVR has stabililty settings to determine how well it responds to transients in the system, and most power factor controllers also have some dynamic adjustments to determine their response capabilities. In my experience most VAR/PF control systems get setup during periods of stable operation, then when system disruptions or transients occur, the VAR/PF controller may get into a dynamic conflict with the AVR, causing some instability, usually seen as large VAR swings.
So I guess if you want to be sure your system is functioning correctly, I'd try monitoring the field output of the AVR along with the other parameters you have. Then you can see the actual system response to some of the apparent disturbances your data showed.
Hope that helps, Mike