During the chazan's repetition of the amidah, do the people in the minyan need to stand? (for parts other than kedusha and modim de'rabanan).
The Shulchan Aruch doesn't say so, but see Rabbi Eliezer Melamed who says that it is preferable to stand, because the rest are considered "partners" with the chazzan in prayer. However, it is not obligatory.
He relies on the Mishnah Berura (124, 20) who says that the tzibbur, by listening and lechaven, are semi-praying as well. And he also says this is an old minhag ("מנהג קדמונים"), and he adds that one may not sit near the chazzan (within four amot).
Rabbi Melamed also mentions the Kaf Ha'Chayim (124, 24), who mentions some more sources that mostly side with standing.
It was his practice to stand with his feet together through the entire repetition of the Amida.
In the Rosh Hashono Machzor “Mesoras Harav” , the section on his hanhagos explains that the Rambam in Hilchos Tefiloh 8 (4) refers to the repetition as “the prayer of the congregation” which means that it is essentially a second Amida. Rav Soloveitchik held that a person's posture should reflect that and that is why he stood with his feet together through the entire repetition of the Amida.
I see this is also discussed in the answer to Why do we still do chazarat hashatz? .