No, in my experience there should not be a natural gas smell from a free standing heater. Natural gas is actually odorless and a scent is added to it just for for this purpose - so that you'll know there is a leak. Free standing heaters are kind of a pain really, and hopefully yours is "tiro balanceado" - the type that is on an outside wall and vented to the outside. People here are somewhat terrified of gas, so erring on the side of caution, it could be because some crappy unsafe appliances are out there in the market. Many people I know here refuse to sleep without first turning off all the gas shutoffs by each appliance.
Some things to try or watch for
Smell of natural gas (very distinctive smell like turning on a gas burner without lighting it). Leak somewhere before the stove or inside the stove or incomplete ignition along the burner. If your feeling bold you can check around the connections with a match, though many will tell you it's unsafe to do so. Spraying connections with soapy water will also help you find a leak.
Flame color: Flame should be a steady blue color. Yellow flame indicates poor combustion and likely means more carbon monoxide is being released. Sometimes you can adjust the flame and get a better color. Properly installed venting also helps.
Apartment humidity is way too high- mold problems etc. Stove could be a contributing factor if not properly vented. Counter-intuitively, water vapor is one of the byproducts of natural gas combustion.
Groggy feeling, headaches, etc - who knows. The damn thing is sucking up all the oxygen or maybe you're getting some mild carbon monoxide poisoning. Either way it's time to crack open a window or two, even if it's freezing outside.