I was pretty giddy and had a lot of mental momentum before. It's a little more complicated than just tailoring the band gaps. Ideally, we would be able to push the band gap anywhere we want, and we might be able to with specialized doping and electric fields, but at the moment we haven't been able to get outside of the IR spectrum (at least in Graphene, which I know a bit more about). There may be some fundamental material property limits that we are unable to get around (I don't think this is the case, but it could happen).
Because Silicon is similar to carbon, Silicon can do many of the things Carbon can. We know Silicon extremely, extremely well. We have been working out the kinks in it for 30 years, Intel has put billions of dollars into researching the material. We know how it works. Other materials fall off very quickly in our level of knowledge, we are still at the point in some of them that we know they work, but we don't know why. Carbon is getting a lot of attention, maybe because it's the next best step, or maybe because understanding Carbon will let us continue to push Silicon forward until years in the future when we have a better understanding of the material when we can transition nicely.
As for making heat -> You could, but the peak wavelength from the black body spectrum of the room would be about 10 microns. These solar cells are significantly shorter wavelength than that (they're absorbing ~1 micron radiation). Also, the longer the wavelength the less energy you get per photon, which wouldn't fare very well.
For heat to electricity, you need a TEG (thermoelectric generators, peltiers, something that uses the Seebeck effect) or a sterling engine of some kind. These work off temperature differentials. Unfortunately, thermodynamics is not nice about recouping energy, and they need high temperature differentials to function efficiently. Even then, the efficiency maxes at about 30% (Carnot efficiency for heat engines).