Can PbCO3 + AgNO3 yield any product? Or do these two not react? If they don't react, can you please explain why? If they do, what is/are the product(s) and the balanced equation?

They don't react as solids and probably won't react (at least no more than a few molecules here and there) mostly because PbCO3 is not soluble. No ions, no reaction. After some time there may be a mixture of Ag2CO3 and PbCO3. Ag2CO3 is more soluble than PbCO3 by a factor of about 1000; therefore, the PbCO3 will predominate and there is no driving force to convert from the PbCO3 to Ag2CO3.