You can run your own test on the freezer while at home. Run the freezer till it reaches desired temperature, then simply put in a cup of water and let it freeze up real good.
Then test your theory. Run the freezer one hour and then unplug it. Wait 12 hours and see if the water is thawed. If it is, then you have to run the freezer more often. Test for a few days in a row until you reach that spot where the water stays frozen between "on" times.
Of course, the hotter the ambient temperature around the freezer, the more it has to run.
Solar would work provided you have enough to keep batteries charged and working.
The longer you run the water taking a shower will be directly proportional to how many showers you can take on a given amount of water. We practice "navy"showers when camping. It's just someting we do. We get 8 showers out of 25 gallons of water this way.