Inverter AC - Doubts - Compressor not shutting off

Though I am not sure about what can be causing your AC compressor not shutting down, here I go with my views on your second question.

If you want to save electricity, the best mode could be Power saver mode. In the auto mode, or Cool mode (if your system has it), the compressor shuts off when a pre set temperature is achieved. But the fan keeps running. The compressor will start again if the temperature begins to rise.
The power saver mode will shut down the fan too along with the compressor. The compressor will not start until the fan has started and the compressor detects a rise in temperature. There is definitely a time gap between the the fan getting started and thereby making the compressor to start, you will save much energy.

However, please note that with the fan getting started and switched off again and again , the resulting sound can disturb your sleep. It solely depends on what your preferences are.

Here are some of the answers to your questions. Do note that your technician and call center were right about the answers they have given you.

1. In case of inverter compressor, it remains on in the background even during the down time. This is quite common for most of the modern AC inverter machines. So the information given to you by call center and technicians were correct.

2. Depending on the temperature around you, it helps if you decide which is the cool temperature. So don't set it to the auto mode. Instead find the coolest temperature required for your room. Auto mode usually tries to adjust itself constantly to the coolest possible temperature. This may not be always good for the room. In such case you have to do it manually.

AC Shuts off whenever the temperature is properly maintained in the auto mode. It also turns itself on when the temperature gets up again. You can use the fan mode when you don't want to turn the temperature of the room too low.

The Inverter AC would be always on working mode unless you manually switch it off. So, to make it work, the compressor has to run continuously.

If you choose the auto mode, then the machine is continuously adjusting itself to the room temperature, to keep it cool, which means it is performing some continual processing which cannot result in power saving.

For power saving you have to choose only the "power saving mode".

If you are manually choosing the temperature, then the auto mode will not work.

The compressor will not stop working even in this case because it is an inverter AC.

Fan mode is generally used, when the room temperature goes too cold, which you don't want and at the same time want some ventilation without shutting down the AC and opening the windows, so that you can switch it back to the AC mode whenever you want.