Brachos 29a - Nine Brachos On Rosh Hashana

The gemara says that the nine brachos we say on Rosh Hashana correspond to the nine times Chana mentioned Hashem's name in her tfilla עלץ לבי [as explains Rashi, that we read about in Shmuel Aleph perek beis] that she said thanking Hashem for the birth of Shmuel. What does that have to do with Rosh Hashana? The gemara continues by explaining that the connection is that on Rosh Hashana, Chana was "remembered" [נפקדה]  by Hashem.

This gemara is PUZZLING! The parsha of עלץ לבי was said by Chana when she went to the Mishkan during one of the three regalim as the pasuk there says explicitly that she went up for the regel with her husband Elkana and Shmuel and the gemara in Chagiga learns the halachos of a kotton and aliya la-regel from that parsha. So how can the gemara learn from here anything about tfillas Rosh Hashana when the tefilla was said at a different time - even if the tfilla was said as thanks for the פקידה that occurred on Rosh Hashana?   

Another problem: According to the Radak in the name of the medrash, the tfilla that Chana said for the birth of Shmuel occured on Shavuos [see Marbeh Bracha קנו] so how can we say נפקדה בראש השנה?

Rashi explains that נפקדה means that it was decreed upon her to give birth on Rosh Hashana. Very nice. But EVERY BIRTH is decreed on Rosh Hashana according to Chazal [Rosh Hashana 8a/16a] so what is different about this פקידה that we have a special memorial on Rosh Hashana?

What is the significance of the number "nine"?

See the preface of the sefer Marbeh Bracha and the Ayn Ayah in Brachos 29a