The largest power plant generates 4GW, not 4MW, as pointed out by @EricDuminil.

I agree with @alephzero. The temperature at the surface drops 10C in 12 hours without sunlight. During the next 12 hours, it will drop less, but still drop a considerable amount. It would cool by 50C within a week and 100C within a month. The problem with this answer is that it fails to take into consideration the speed of the heat transfer to the surface. So after a month, the surface might be 100C cooler than it was, but 20 meters underground, the temperature might have only fallen by 20C or even less.

Your water requirements have a serious math error, although it doesn't impact the energy costs much. An acre has 43560 square feet. 125mm of rain is 5 inches. There are 7.48 gallons per cubic foot of water. Let's just say that one gallon = 4 liters. So the water required is 43560 sq feet * 5/12 cubic foot of water / sq ft * 7.48 gallons / cubic foot of water * 4 liters / gallon. This works out to 543048 liters of water. Further, 5 inches is the bare minimum to get a crop. Above that, yield increases by about 7% per additional inch of rain. Typically 16 inches would be a floor.

500 Million being the limit is a political statement, not a technological one.

You can't have a tiny neutron star. What makes a neutron star is all that mass being concentrated in a fairly small area. If you remove a small piece from the star and place it at a distance from the star, that piece would expand greatly because it would not feel the gravity from all the mass that used to be near it.

Regardless of the materials involved the net solution has to be based in space. If the net is strong enough and massive enough then it can absorb the energy of the impactor, and convert the combined speed of the impactor plus the net to a much lower velocity. If the net were high enough and not in orbit around the planet, then after the impact the combined net and impactor would not be moving as fast as the planet and the net would not impact the planet at all. Consider a net perhaps 100 times the mass of the impactor. It's not very practical, but it implements what you are asking about.

Of course humans would survive, there would only be 5000 people to compete for all the world's resources. There would be quite a few deaths in the first year. Two factors would determine how quickly technology could be regained: 1)The distribution of the population. If the people are widely scattered then regaining technology would be a very long process. 2)The quality of the initial leadership. Leadership needs to understand that besides surviving, the most important the 5000 could do would be to preserve their knowledge for future generations.They don't need to restore technology.

This answer is very good overall. It ignores the effect of ocean currents, which would be substantial. The dark side would be colder than the number stated. The warm side would be substantially less warm than the speculation due to thermal radiation being proportional to the fourth power of the temperature. Also there would be some planet-wide convection. Not a lot, but enough to moderate the temperatures some. Also the ice that would form in the ocean at night would help moderate the daytime temperatures to some extent.