What is/are the earliest origins of the customs to stand during various passages of the daily service before the Amidah (e.g. Baruch She'amar, Shirat Yam, Yishtabach)?
Partial answer regarding Az Yashir. It seems that the minhag to stand for this is quite old. From here :
In many kehilos the minhag is to stand during the aliyah of Shiras HaYam from “Vayosha” until the end of the Shirah (Sefer Ketzos HaShulchan 84, Badei HaShulchan 22). One reason is based on the idea that the recital of the shirah by Moshe and Bnei Yisroel was comparable to the recital of Hallel (Mishnah Sotah 27b). The halacha is that Hallel is to be said standing (Shulchan Aruch 422:7), because one is testifying to the fact that Hashem did miracles for us, and testimony must be said while standing. Therefore, the custom is to stand during the shirah, and perhaps this is also the reason why some people stand for Az Yashir during pesukei d’zimra (Kitzur Shulchan Aruch 14:4; Badei HaShulchan ibid.). Another reason for standing during the shirah is based on the Zohar (Lech-Lecha 81b) which says that Dovid HaMelech merited to be the ancestor of Moshiach because he stood up in order to say shirah, as it says (Tehillim 119:62), “I will arise to praise You” (Siddur Tzelosa d’Avraham, pg. 168).