What is the practical application of 'you should follow a proper Beis Din'?

Bechor Shor: It is a Mitzvah to go to a more authoritative Beis Din outside your city, even if there is already a functioning Beis Din in your own city.

What was the message behind the millstones and the candles?

Rashi, R' Chananel, Teshuvos haGeonim: Since the Romans banned circumcisions and weddings, the local community found discreet ways to spread news of these events without drawing the authorities' attention. (The Yerushalmi in Kesuvos 1:5 adds that even after the ban was released, these customs continued.)

What is the meaning of the phrase 'Shavua haBen'?

Rashi: It refers to the Bris Milah, as a week passes before a boy is circumcised on the eighth day.

Maharitz Gias (Hilchos Avel p. 212): They would have a week ('Shavua') of celebrations each evening upon the birth of a boy or girl. (This is mentioned in Maseches Semachos 2:3.)

Why is 'Rome' part of a list of cities which are in (or relatively close to) Eretz Yisrael?

Margalios haYam: The city called 'Romi' is not Rome in Italy. It is a town in Eretz Yisrael, near Tzipori in the Galil.

Why are the Chachamim in the Lishchas haGazis the last in the list?

Yaavetz: To teach that even if there are greater Chachamim elsewhere, one should still go to the Sanhedrin in the Lishchas haGazis, because the unique authority of its rulings comes from the place itself, the Beis HaMikdash, giving the rulings extra weight beyond just the wisdom of the Chachamim who sit there.