Overheating and no heat. No coolant in the oil, new thermostat, water pump is good, coolant is full, both lines to the radiator are cold. What could be the problem?
The coolant in the engine cooling system requires it to be circulated from the radiator to the water pump, thermostat, engine, and then back to the radiator. If the radiator hoses to and from the radiator are cold, then you do not have any circulation through the radiator. Another possibility is that the water pump impeller is worn off and not circulating. This is why the heater is not working since there is no flow through the heater core.
Your radiator and heater core may be clogged and would need to be checked by draining and disconnecting the lower radiator hose, and then applying water through the radiator top where cap goes to see if water flows through the radiator. If the water does not flow through the radiator, then check the heater core for flow also. If both are clogged then replace them.
If both flow okay then the water pump is bad and has no fins to circulate the coolant throughout the cooling system. If you need help diagnosing the problem then contact a mechanic like one from YourMechanic, to come diagnose and fix your overheating problem for you.