What is the standard molar weight of Carbon, and why?
As you stated the molar mass is a weighted average based on isotopic abundance, specifically abundance on Earth. We can obtain the molar mass of carbon by averaging the mass of $\ce{^{12}C}$ (about $98.938\%$ abundance) and $\ce{^{13}C}$ (about $1.078\%$ abundance). $$(\mathrm{mass\space of\space \ce{^{12}C}})*(.9889)+(\mathrm{mass\space of\space \ce{^{13}C}})*(.0111)=(12)*(.9898)+(13.00335)*(.0111)=12.0206\text{g}$$ This deviates from your value above because the values I used for the abundance aren't very precise.

In terms of where these abundances comes from and how the error bars are determined, the short answer is that its complicated. The long answers is here in a technical report by IUPAC, particularly sections 1.2-1.4.