1. In the time of the Beis ha'Mikdash, the two days fell on the 30th of Elul and the first of Tishrei respectively because the real Rosh Hashanah was only fixed on the second day, as the Gemara explains.

Nowadays, when we know for a fact that, according to the calendar, Rosh Hashanah falls on the first day (which is of course, the thirtieth of Tishrei), it would be pointless to postpone the main day of Rosh Hashanah until the second day.

Consequently, we end up commemorating the two days of Rosh Hashanah that they celebrated then, on the one hand; but on the other, we count the first day as the first of Tishrei, despite the fact that they counted the second.

2. The Gemara at the foot of the Amud answers your question. According to Rashi, it is because, even after Raban Yochanan ben Zakai rescinded the decree not to accept witnesses after Minchah, they continued to observe the second day. Whereas according to Tosfos, who disagrees with Rashi, they would still keep two days if the witnesses did not appear until the next day. Notice, that in Rashi's case, the first day would occur on the first of Tishrei (and not on the thirtieth of Elul) like your first question.

Be'Virchas Kol Tuv.

Eliezer Chrysler