Q: Would a bad pcv valve make the engine run hot

Hi there. From the description you have provided, your vehicle’s bad PCV and coolant loss are not related. The PCV (positive crankcase ventilation) valve is designed to remove combustion gases that get past the piston rings into the engine. Typically, a bad PCV valve leaks or plugs, replacement is recommended. The coolant/antifreeze loss, is a leak in the engine’s cooling system that requires repair sooner than later. Any number of components in the cooling system could be leaking, ie; water pump, radiator, cooling system hoses. A cooling system pressure test could confirm the area of the leak. Overheating the engine could result in a catastrophic and expensive failure. I recommend having your vehicle’s bad PCV valve and engine runs hot be diagnosed and repaired by a certified technician, such as one from YourMechanic.