How did we get the whole world to agree to the same calendar/year system? And how long did it take for people to start thinking "right now we're in year ____"

I can't answer the whole question, unfortunately, but I can say that the whole world really hasn't agreed to use the same calendar yet.

For instance, I live in Taiwan (Republic of China). The year in Taiwan is 103. That is, 103 years ago the Qing dynasty fell and the Republic was formed, and on all official papers and notices you will see the dates like such : 29/06/103. (The 29th of the 6th month of the 103rd year.)

Most people are aware that according to The West this is the year 2014, in the same way Europeans are aware that America uses feet and not meters, and when doing international trade the Western calendar is almost exclusively used. But within Taiwan, it is the year 103.