Whether you can continue using your old power supply depends on three factors.
All the necessary information to calculate it can be found on the back of the power supply. To calculate the output power of the unit, use the specifications listed under "output".
Calculation:
Ampere * Volt = Watt
Example: 4,74 A * 19 V = 90,06 W
The power supply therefor has 90 Watt.
Yes, you can use a power supply with a higher wattage, however you need to pay attention that the voltage doesn't deviate more than one volt from the voltage of the original power supply.
It also doesn't work the other way around; your new power supply needs to have at least the same wattage as the old one.
To use your laptop in a vehicle you need a car-adapter, which converts the 12 Volt your car provides to 19 Volt most Laptops need.
If there is no car-adapter, that is compatible to your notebook, you can use an inverter which generates mains power from the power, supplied by the car. You can then plug your normal power supply into it.
Not necessarily, try taking your battery out (if possible) and starting the computer while connected to the charger in question. If it starts, the power supply is working but something in the Notebook is broken.
There are also multiple indicator lights, on the laptop, whether it is being charged, also this can be seen in most operating systems. You can also use the indicator light found on some chargers to diagnose an issue with a cable.
Power supplies are subject to certain specifications and can reach up to 50°C. The most amount of heat is produced when working on the Notebook and charging the battery at the same time. As long as you can touch your charger without getting burnt, its temperature is probably within specification. However, it is most important, that the heat can be dissipated, therefor you should never put your power supply under a blanket or in a closed drawer.
No, because a circuit in the Notebook controls how much power arrives at the battery and not the charger.