Is it programmatically possible to communicate with Siri via bluetooth? We see people talk on bluetooth headsets all the time. In addition bluetooth is used to send notifications to devices like watches. 

My question is Is it programmatically possible to communicate with Siri via bluetooth?
Yes, some Bluetooth headphones, speakers and hands-free car audio systems have Siri functionality, the Bluetooth device acts as a speaker/mic with action buttons, one commonly found being the Call button.

So, in answer to your question, yes it's possible with the right equipment.