Q1: What is the earliest source for blowing at the end of Ne'ilah?
Q2: What does the Te'kiah accomplish?
Q3: When is it sounded?
Q4: Which notes ares sounded?
A1: I'm still not certain, but it seems to be post Talmudic
Several reasons are "proferred" in B"Y
A. Related to Yoveil - Mordechai
B. So as to co-incide with silluk haschinah. - Hagahot Maimoniyyot
C. Tosafot - to indicate it's nightfall. Arguably a good reason to blow after night "mamash" and perhaps to delay until after Arvit
A3.Several options are mentioned:
A. Yeish Nohagin Miyydad after Ne'ila
B. Some before "sheimos"
Bottom line, there is not a rigid consensus at the end of the Rishonim as to how this works.
EG The Bach does favor as close to Ne'ilah possible, but this appears to be a preference not an absolute because he characterizes it as "adif t'fei"