Sensors have a native ISO
No they do not. ISO is an output format (e.g. JPG) metric defined by ISO 12232 standard. It has nothing to do with image sensors.
Most cameras do change the operational parameters of the image sensor when the ISO setting is changed, typically the PGA (programmable gain amplifier) setting is changed and the signal is amplified the more the higher the camera’s ISO setting is.
and any extended (higher or lower) are simply the sensor amplifying or decreasing the signal.
That is not right.
Typically the ISO 100 (sometimes something else) setting on a camera is such that the image sensor is run at the lowest PGA amplification setting. Anything above that and the amplification is increased (and/or digital multiplication is used in software). The “extended lo” settings typically operate the sensor at the very same setting the ISO 100 does, just change the metering and processing of the data.
Grain is only one part of the noise function. The other is the random nature of light itself.
Nope. We can of course define “sensitivity” is multiple ways, but typically in this context the image sensors have one sensitivity (though different for each wavelength of light).
No, this is plan wrong.
Here’s what happens typically:
The reason why there is step 3 is in step 4 noise. If you increase the signal before ADC, the noise that the ADC adds becomes less relevant in comparison. The drawback for amplification is that the signal may “burn”, or the ADC operational range is exceeded.
A fun experiment is that you take two shots with identical exposure settings: one at ISO 100, the other at ISO 6400, both in raw format, and then process them in Lightroom or other such raw processor to the same lightness. For practically all today’s cameras the ISO 6400 will appear cleaner.