A portable stove work a littel bt different depending on the type, if it uses gas o just electricity. But in general the structure of the enery process is the same.
1. The portable stove has a source of energy (gas or batery).This source of energy is the first state of the energy and have enough to start and cook.
2. When your start, the energy have to trabnsform into heat so you are able to cook.
If it has gas, then the fire starts and the heat is generated by burning the gas.
If it is electric and has a battery, the electric energy goes into a resistance that increase the temperature.
3. With heat you are able to cook what you need, having in mind that the limit is given by the initial source of energy.
Now lets answer the questions:
Q1. The law of conservation states that the total sum of energies of a systems remains constant always. While you are burning gas or using the resistance, the heat energy plus the looses of energy have to be the same as the amount of energy you spend from the sources, batteries or gas. With that idea you can calculate how many heat you can produce based on the source.
Q2. They prove it by comparing them. If you compare the initial what you spend with heat and losses you would get the exact same result.
Q3. All power plants have energy in different ways that we can't use, kinetic energy, solar energy, heat and others. Then the have to transform it into electric energy to pass it to houes, factories and other places. This transformation process might chenge depending on the source you are using.