Why did Turnus Rufus's question seem to focus on this one day rather than two days (which is indicated by the 'Yomayim')?

Ben Yehoyada: Nowadays, there are two world religions that observe Sunday and Friday as their Sabbath - Christianity observes Sunday and Islam observes Friday. It could be that even in Mishnaic times, those two days were also observed. Turnus Rufus was asking why the Jews observe Saturday as Shabbos.

Why did R. Akiva brings three proofs as to which day was Shabbos - the Sambatyon river, a necromancer, and the grave of Turnus Rufus's father?

Maharsha: The Maharsha explains that Turnus Rufus's challenge actually consisted of three separate questions: Who says that the seventh day is the day on which Hash-m rested after Creation? Who says that this is the day on which Klal Yisrael was commanded to refrain from Melachah? Who says that Shabbos will remain a day of rest even in the future, including in the World to Come? R. Akiva therefore responded with three corresponding proofs: The Sambatyon River, which flows turbulently throughout the week but becomes calm on Shabbos, demonstrates that this is the day on which Hash-m rested, since the river has followed this pattern since Creation. The necromancer, who is unable to summon the spirits of the dead on Shabbos, demonstrates that this day is prohibited for Melachah. The grave of Turnus Rufus's father, from which smoke ceased to rise on Shabbos, demonstrates that Shabbos remains a day of rest even in the next world, as he was not subjected to punishment on Shabbos.

Iyun Yaakov: Three proofs correspond to the three regions of the world - the inhabited area, the sea and the desert. The Sambatyon river is a proof from the sea; the Ba'al Ov (necromancer) is a proof from the inhabited area and his father's grave is a proof from the desert.

What is the meaning behind the three terms (shamed, embarrassed, cursed) that Turnus Rufus responded with and who was he speaking about?

Ben Yehoyada: He was speaking about himself. 'You have shamed him' - by comparing him to regular men; 'you have embarrassed him' - with your successful proofs against him; 'you have cursed him' - by saying that smoke rises from his father's grave.

Why did these Chachamim create a calf on Erev Shabbos rather than on other days of the year when perhaps there was an even greater need for meat? (Rashba)

Rashba (Teshuvos 1:413): The Gemara in Taanis 27b describes the prayers of the Anshei Ma'amad, who represented the Jewish people while the Korbanos were offered. Their prayers varied according to the day of the week because whatever was created on a particular day of Creation is stronger on that same weekday. Thus, they prayed for sailors on Monday, when the seas were created and for desert travelers on Tuesday, when dry land was formed. Likewise, since animals were created on Erev Shabbos, the power to create them is strongest on that day. Therefore, R. Chanina and R. Hoshayah specifically created a calf on Erev Shabbos.

Toras Chaim: Creating a calf through miraculous means was justified only because it was done in honor of Shabbos; otherwise, it would have been inappropriate. This is illustrated by the Gemara in Taanis 24a: When R. Yosi of Yokras was delayed, his hungry workers asked his son for food. The son miraculously caused a fig tree to produce fruit before its season to feed them. R. Yosi rebuked him for troubling Hash-m to perform an unnecessary miracle, and his son died young. Similarly, creating a calf simply for meat would have been inappropriate, but doing so in honor of Shabbos justified it.