Can a home owner rightfully have cars towed away that are parked on the street in front of their house even though there are no "No Parking" signs?

NO. Even if you park two of your tires on there lawn they cannot do anything about it.

I wouldn't think so. It would depend on the laws of your city/township. You should check with the city or the impound lot to see why they were called in. Some people get upset if the car is parked in their carriage walk. If you have an open communication with your neighbor you could ask what the problems is?

Where I live if a car has been sitting in the same spot for over 72 hours it can be ticketed and then if still not moved it can be towed away. Towing prior to that can only be done if the car has been stolen or is unregistered or has expired tags.
Best of luck