Is use of propane gas for heating and cooking hazardous to human health?

That is not the case and many, many, years of use has borne this out. As with natural gas there is always risk of explosions from a leak that gets ignited, as well as potential asphyxiation, and prior to that any number of physiological impairments due to lack of oxygen if too much air is replaced by the propane. But trace amounts are of no consequence for humans to breathe it in, or for the environment.