I have not heard of a sand dog element. But with hard water you can put in a stainless element that will not burn out. Most of the times the lower element will burn out because of build up that forms in the bottom of the tank. This is why some people will drain off part of the tank every few months to try and flush it out. Where I live (very hard water) I say it should be a monthly thing.
As far as energy savings to heat water it will cost the same no matter what wattage or type of element you use. You use the same amount of watts to do the same amount of work. But if there is a build up of deposits or a bad dip tube it will take longer to heat the water so it will cost more. The water temp is up to you. With kids I would go with the lower setting.
As far as the water temp IMO the cost in hot water is to heat the water. Once it is hot and no water is being used I don’t think it makes a difference on the energy bill to keep it hot (this is based on the newer heaters; but not in a cold unheated room). If you don’t keep up with flushing the tank then you will run into more problems in down the road.
Depending on how old it was and if you can get most of the build up out of the heater I would replace the upper and lower thermostats and elements. I would not get any high $$ top of the line “sandhog” or stainless elements.
I’ll try to find a post on flushing the heater and edit it in for you.
This post has some good info on the build up and how to clean them out.