Originally bais din would accept witnesses that they saw the new moon all day. One time on the 30^th of Elul bais din was waiting for the witnesses to come, they didn’t so they brought the korban tamid of the afternoon and the leviim didn’t know what song to say. They had a question to say the weekdays as the witnesses didn’t come or the song of Rosh Hashana if they would come.

To get out of this question the chachamim said we will not accept witnesses only till the time of mincha. If the witnesses come afterward, instead of just that day being kodesh they said that day plus the next would be Rosh Hashana.

When the bais hamikdash was destroyed that bais din can accept witnesses the whole day as there is no more a problem with the song of the leviim.

R’ Yehoshua ben Karcha said R’ Yochanan ben Zakai instituted that witnesses that come and say the new moon can only come to the place were bais din normally sits and not anywhere where even the head of the bais din is.

Rosh Hashana 4:5

The prayer of Mussaf on Rosh Hashana is built upon the blowing of the shofar plus saying malchiyos [kingship] zichronos [remembrance] and shofaros [ blowing the horn].

In mussaf first you say the brachos Avos [fathers] Gevuros [ strenght] as usual then kedushas Hashem [holiness] plus malchiyos without blowing the shofar. Then the holiness of the day with blowing the shofar, zichronos with blowing the shofar, shofaros with blowing the shofar. Then you go back to regular and say Avoda, Hodaa and the blessing of the cohanim. These are the words of R’ Yochanan ben Nuri.

R’ Akiva says you say malchiyos with the holiness of the day and blow the shofar.