Not a guru by any means, but your flow chart shows very logically what you want and the Arduino certainly has the means to execute the logic (enough memory, enough speed, etc). The Arduino would seem to have enough inputs and outputs to handle the tasks also. What is not clear is how much external circuitry will be required to interface the various inputs and outputs.
Switching audio sources between the various sound generators (radio, mic, ipod) to the one headset is pretty simple and can be done with either Arduino controlled relays or op-amps (I think headphone audio might be too high a level and current to use simple analog mux chips by themselves).
You will need some kind of analog op-amp circuits to be able to detect the presence of sound from the mic and radio so that adds to the external circuitry needed. Mic level is especially low level so that would take some extra amplification, radio sound output is a little higher output so possible a simple diode rectifier/filter circuit would drive a analog or possibly even a digital input to the Arduino.
Lastly you need to be able to control the ipod. That I have no knowledge of, so the first question is does the ipod have external control inputs that would allow manipulating the ipod operations? If not it would be a most difficult problem. if it does you would probably just need external interface components like optical isolators to command the ipod.
So the big challenge is not in the Arduino, the real challenge is all the external interfacing circuitry that you will have to develop. And that would be true no matter what brand or model of micro controller you try and use.