When exiting the US should i give the airline my other passport (knowing that i am flying back to my other country of citizenship) or should i give them my US passport?
Give them your other passport. That's what I always do.
Will the self-check in Kiosks detect that my US passport has expired?
Not if you don't use it in the self-check in kiosk. If you do, it will reject the passport because it is expired.
If i use my other passport to exit the US, will the Kiosks detect that i entered the US with my US passport and am not allowed to exit it with another passport?
No. If you flew in with the same airline, they might notice that you checked in for the incoming flight with an expired US passport, but there is no rule against using a different passport to leave.
What are the risks?
It is "unlawful" for a US citizen to leave the US without a valid US passport. There is no penalty for doing so, however.
I have fairly extensive experience leaving the US on European airlines with an EU passport after having entered with a US passport. Nobody ever mentioned that I hadn't used the EU passport to enter. Nobody asked about my immigration status (though it's a VWP-eligible passport).
There is an extremely small chance that you might encounter a CBP officer screening departing passengers. Such an officer might determine that you are a US citizen who is leaving without a valid passport and prevent you from boarding the plane. (I've never encountered such a screening myself. They are uncommon in general, but I have read that they are fairly common for flights to certain destinations.)
More likely, a US airline might be more interested in the immigration status of departing foreigners than are European airlines. I've read a comment on this site from someone with dual nationality who was challenged when checking in for a flight leaving the US using a passport other than his US one; this was, however, with a US airline, so that is likely why his experience was different from mine. Also, an automated kiosk once required my wife to scan her US visa when checking in for a Delta flight leaving the US. (Her passport was not from a VWP-eligible country; I was using my US passport so I don't know what would have happened had I tried to use my other one.) The airline might have a policy of requiring US citizens who check in with a non-US passport to have a valid US passport on departure. I have never encountered such a policy, however.