Cost of liquid propane (that's what the LP stands for) vs. electricity depends on various factors. We are happy when it's warm enough for our heat pump to run because it's cheaper to heat with electricity than propane, but we are on off-peak rates for the heat pump and we have low rates here to begin with.
Liquid propane vaporizes at temps warmer than around 45 below zero. The gas is what's burned off to heat a house or light a stove burner. The tanks are pressurized to force the gas through the pipe to the house.
Propane appliances are often more efficient. Our camper fridge works better on propane than on electricity.
I'm not sure residential propane fridges are the gas absorption type like RV fridges or if they use a compressor. If they are gas absorption, they only cool to about forty degrees below ambient temp. So if it's ninety degrees in the room where the fridge is, it would only cool to about fifty degrees, which is not safe for food storage. I know a lot of people in our area use gas fridges because some of the more remote cabins around here are so far off the beaten path and it's prohibitively expensive to put in power lines. They use gas lamps and other appliances, too.
As for solar power, how well that's going to work depends on where you live. Here, winters are usually quite cloudy for long periods of time and days are short. Solar wouldn't work as well for us as somewhere the sun shines shines almost every day. You can do a cheap experiment by buying a small solar garden light and putting it in a sunny spot where you can see it easily from your house. Keep track of how many nights that light glows brightly and note what the weather conditions are, especially during winter. Here, it's not unusual to get many nights in a row when the lights aren't on at all or are quite dim. That would give you an idea how well solar power might work in your particular location.
