We usually don't remove the petcock because they're plastic and they break, and often the plastic radiator tank cracks and you'll have a slow leak. The engineers know this. It has been a problem since the early '90s but it's good for business.
Some petcocks push straight in, then you turn them a quarter turn clockwise. If yours pops back out you may not be pushing it in far enough. Also compare the new one to the old one. If nothing looks broken on the old one, the pieces you found are likely from inside the radiator and it will have to be replaced.
If a small piece fell inside the radiator, that is not going to cause overheating or any other problem. The overheating is most likely due to the system needs to have an air bubble burped out. That air pools under the thermostat. Thermostats open in response to hot liquid, not hot air. Look for a bleeder screw on the thermostat housing at the end of the upper radiator hose. If there isn't one, look for a threaded plug or a coolant temperature sensor that can be unscrewed. Fill the cooling system with that item removed. If you unplug a sensor to do that, do it with the ignition switch off to avoid setting a nuisance diagnostic trouble code.
Thursday, June 6th, 2013 AT 1:32 PM