Hello Angel 3.
Over the years our caravans have become more like a home extension, and we have come to accept the convenience of mains power. So to be without mains power means we have to be far more frugal with the way we use the battery.
Lets answer the simple question first, Yes provided the car has been wired up correctly, then when you tow the car will try to recharge the battery. If you have any doubt about the car to battery charging, then your local caravan dealer or tow-bar specialist should be able to check it for you quickly and easily. Check your instructions for the caravans switches to see how they should be set.
The fridge should be perfectly happy using gas, (and incidentally when towing the fridge should also be cooled by power from the car see your instructions for the fridge.
Charging phones, this could be done in the car. If you haven't already got a car phone charger, the're not very expensive and easily available. Solar chargers could be an option.
Now for some tips on power management:
Take the biggest battery you can fit into the battery box. This will most likely be a 110Ah battery. 110Ah describes the capacity of the battery, and in theory means you could run a 1A device for 110Hours, or a 5A device for 22 hours and so on. In practice its not quite that simple because as batteries age, they loose some of their ability to accept and hold a charge, and for safety reasons the caravans built in charger will only charge a battery to about 85% of its total capacity. If you can, charge your battery at home before you go away with multistage charger to make sure its fully topped up.
When in the caravan, only turn on lights and appliances when you need them, and possibly more importunity turn the off after use.
Lights are one of the devices that are deceptive to the unwary. Most conventional caravan lights draw about 1A to 1.5A whilst there on. It doesn't sound much, but because lights tend to be used for long periods it soon adds up. Even better change the lights for LED models which consume much less power, but that is quite an expense.
Space heaters, This will depend on the make and model of heater you have, but I will assume you have a Carver or Truma convector heater. These use gas to provide the heat, but most models have a 12V fan to distribute the hot air. Use the fan as little as possible because like the lights it uses about 1A of current and would normally be left on for long periods.
Water Heaters, Again I will assume you have either a Carver or Truma water heater. These use gas to heat the water, but the need 12V power to control them. They will use about 0.5A when heating but that falls to about 0.1A in stand by. The heaters take about half an hour to heat a tank full of water so you need to predict when you are likely to need them and switch them on in good time. Because they are insulated, the water will keep quite warm for two to three hours, so you don't need to turn them on for washing hands etc during the day.
Other items of caravan equipment such as the water pump tend to have intermittent usage, and although they draw bigger currents, the short time means their actual power usage is not all that big.
Electric kettles, Hair dryers and microwaves on 12V are a No No.
Now for the luxuries. Televisions,computers these tend to have moderate power usage, but they also tend to be used for longer periods, so these will tend to deplete the battery quite a lot. Be very selective about their use, even consider leaving them behind.
You can extend the capacity of the battery if you were to consider getting about a 50W solar panel fitted, or get a second battery, Many sites offer a battery charging service, or by taking appropriate safety measures you could recharge the spare battery in the car whilst your out and about.
Generators are a possibility, but they are noisy (even the "silent" ones) and they are hopelessly inefficient and quite expensive. Many sites limit or even completely ban their use.
Following this sort of strategy four of us have used a caravan for 6 days in the summer without mains.
I hope this helps.