Why is he called Zerubavel?
Sanhedrin 37b: He was Nizra (conceived) in Bavel 1 .
Ibn Ezra: It is due to Bnei Tziyon who are Sheruyim 2 (dwelling) in Bavel.
Sanhedrin 37b: His true name was Nechemyah ben Chachalyah. Ramah - he was the son of Yechanyah! Also, verses list Zerubavel separately from Nechemyah! Margoliyas ha'Yam - 'ben Chachalyah' means 'the son of one who was Chayav Kelayah (worthy to be destroyed).' A different Nechemyah was listed with Zerubavel.
This is close to Zeruyim (scattered). Perhaps Zayin and Shin are interchangeable, for both of them are pronounced from the teeth (Maharsha Berachos 6b). (PF)
Zerubavel was not She'alti'el's son, rather, his nephew!
Ibn Ezra: Because he raised him, he is called his father. So it says [that he was his nephew] in Divrei ha'Yamim I, 3:17-19 "u'Vnei Yechanyah Asir She'alti'el Bno; u'Malkiram u'Fdayah...; u'Vnei Fedayah Zerubavel..." We find that Bnei Merav bas Sha'ul are called Bnei Michal (Shmuel II, 21:8) because she raised them.
Did they build the Mizbe'ach only to offer Olos?
Malbim (2, 5) #1: Yes. Verses 3-6 support this. This is like the Ra'avad (Hilchos Beis ha'Bechirah Perek 6), who says that Kedushas ha'Mikdash ceased at the Churban. And it is like R. Shimon (Zevachim Perek 14), that on Bamas Tzibur, even the Tzibur can offer only Pesachim and obligations with a fixed time, that an individual can volunteer similar to them. These are Temidim and Olos (Musafim) during the Chag, but not goats (Chata'os) of the Regalim. Also the Avodah of Yom Kipur they did not do, like Tosfos says. All the more so they did not offer obligations of individuals, and Menachos, which are not offered on Bamos. They did not offer Zivchei Shalmei Tzibur (Kivsei Atzeres), for they are Kodshei Kodoshim, unlike Shelamim that individuals may bring.
Malbim (5) #2: According to the opinion that we may offer Korbanos even without a Bayis, we must say, with difficulty, that "Olos" is not precise (they offered also other Korbanos).
