Is my rental house safe even if the electrical is not grounded at all?
I agree with Aloysius. Many homes in the U.S. Are 2 wire when the 3 wire outlets were added each circuit should have been protected by a GFCI and the outlets marked no equipment ground. Other that "touch lamps" I don't remember seeing lights that required the ground to work.  The hot smell is not a good thing make sure you are not overloading the circuit. I would guess that the electrician only checked the polarization of the outlets with a outlet tester like this or one with a GFCI test button.  If the circuits are GFCI protected with a GFCI outlet on the first outlet in the circuit or a GFCI breaker the system would meet code. The second tester cost ~10$ and you could verify the polarity and push the test button to see if there is protection.