Asus 65W AC Adapter Output: 19V DC-3.42A 4PHI

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.

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.