When the new month was determined by Beis Din, what was the rationale for sending messengers to the Diaspora to publicize the date of Rosh Chodesh Elul? How many days of Rosh Hashanah were then observed in the Diaspora?

When the new month was determined by Beis Din and then publicized by messengers, were two days of Rosh Hashanah observed out of doubt in other locations (where the messengers of Tishrei would arrive only later)?

Nowadays, when we have a fixed calendar, how many days of Rosh Hashanah should be observed in Eretz Yisrael?

Are the two days of Rosh Hashanah considered to be one period of sanctity, or two? What are the various opinions as far as Eretz Yisrael vs. the Diaspora, and the period following the Churban vs. today's fixed calendar?

What are some practical ramifications of the question whether the second day of Rosh Hashanah is a continuation of the first day, or has independent sanctity? Is Shehecheyanu recited?