That expectation is not justifiable by the physics. The forces from the applied electric field are quite small compared to the forces from the Coulomb between the water molecules and the water and the ions, yet the time of your observation is extremely short. Even experimentally, the effect of ions diffusing preferentially to differently charged electrodes is quite slow. The magnitude of an electric field needed to observe the behavior you expect would be so large, in a real experiment there would be a flash all all water would be evaporated. You would need average over a much larger system and a very much longer simulation time to see any effect of the kind you describe and realistic choices of electric fields. Most simulation studies use field strengths one or more orders of magnitude larger than what is experimentally viable. Check the corresponding published literature. There is a lot of that. I did that kind of simulation 30 years ago as an assignment when I was an undergrad.
Why? You add forces to a system, that means adding energy. When you add friction to your hands (i.e. rub them) they get warmer, too, don’t they?
Also, you have to first validate that your system that conserve energy without fix efield to make certain, you don’t have an energy drift due to bad simulation settings.
By how much? A small drift is to be expected for simulations employing floating point math and interactions with a cutoff.
You don’t say how large the system is and how high the NaCl concentration.
This is a bit of a bad idea because for fix efield the potential is inconsistent with the force when crossing a periodic boundary. Also you may induce a drift of the center of mass and eventually may become victim of the flying icecube syndrom in case you are also applying a thermostat.
As mentioned above this is not a question you should ask here, but a topic for a search of the published literature. That is the best way to get access to this kind of information. That is why people publish their findings (and to graduate and/or stay gainfully employed, of course).