Skype Phone Project - Help with Audio Amplifiers Selection
You should be able to utilize the phones internal button used to activate the hang up action to disconnect the "sense" function of the USB audio device. ie; tell the computer that you have disconnected the headphones and it should send the audio through the original default device, the desktop speakers. 

You would want to like you said adjust the volume on the phone so that the levels would stay even and comfortable, I would suggest installing a 10k logarithmic potentiometer as a voltage divider to simply cut down the volume going to the handset speaker.

If you are going to be using a button for push to talk you should not have a problem with putting the phone down, it will already be muted when you release the PTT.

There should not be any need for an Arduino here, maybe a couple op-amps if your mic level is to low. 

Sorry I dont have more time to point you to some components.