In general, no, natural gas is not part of our infrastructure and people don't use it for heating. This has to do with our easy access to hydroelectric power.

I don't know about US, but in Europe we have modern woodstoves like this maker: http://www.hase.eu/

Well installed, they burn almost everything, leaving only mineral parts of the wood. That means more energy-efficient burning, and less smoke. In addition since they're completely close all the smoke goes through the chimney, nothing gets in the house.

Correctly used, you get almost no smoke (because everything burns). The only reason a recent woodstove would produce a visible amount of smoke is if you burn stuff you're not supposed to burn (some people use their chimney/woodstove as a garbage bin) or if you burn wet wood.

So even when other energies are available, it's a pretty good way to heat your house: cheap, efficient, nice warm feeling, limited pollution.