In Arduino the clock signal comes from a 16MHz crystal oscilator. If I program the device
with arduino language using it's functions like "delay()" or "tone()", if I put the microcontroller
to operate using the internal RC 8 MHz oscilator as clock source will those functions work
The Arduino core libraries (delay(), millis(), tone()) work properly at 16 MHz and 8 MHz. You have to use the correct Tools->Boards and re-compile the sketch to have it work properly if you are switching the clock frequency.
From AVR clock controll unit comes several clock signals. Which frequency does Clk/cpu has?
Which frequency comes to clk/adc? This is not clear to me in datasheet.
If you look at figure 8-1 (page 26) you see that the clk(ADC) comes out of the AVR Clock Control Unit. This is fed from the System Clock Prescaler so the ADC pre-scaler starts with the system clock frequency. To get the 16 MHz system clock down below the 200 kHz upper limit for full-scale ADC conversions the ADC pre-scaler is set to /128 so: 125 kHz. For an 8 MHz clock you could /64 to also get 125 kHz ADC clock. You can use a smaller ADC pre-scale to get a faster ADC clock but you won't get 10-bit precision.
For signal processing at audio frequencies you might want to use an external SPI ADC.