Why does the Torah see fit to define "Beis ha'Sohar"?
Ramban (citing the Ibn Ezra): Although the word appears to be a regular Lashon ha'Kodesh word, the fact that the Torah sees fit to define it suggests that it is basically an Egyptian word, since it is the way of the Torah to define foreign words. 1
Ramban: The Torah is not defining the word (which is purely Lashon ha'Kodesh), but explaining that Potifar placed Yosef in the prison where the royal prisoners were incarcerated. 2 Or, it was the prison where those who sinned against the king were incarcerated. 3
Ramban Which explains why the chief butler and baker were later incarcerated there, and the Torah mentions it because it raised Yosef's status (See R. Chavel's notes).
Ramban: In spite of which Potifar placed Yosef there - because he loved him.
What exactly, does the word "Sohar" mean?
Ramban #1 (citing the R'dak): It means 'a round arch'. 1
Ramban #2: It refers to a pit - an underground prison, with a small opening on top into which the prisoners were lowered, and which let in a little light. 2
Its source is the Arama'ic word 'Sihara' - the moon (which is round and which illuminates). It is like the word "Tzohar" ("Tzohar Ta'aseh la'Teivah" - 6:16), so-called there because it gave a lot of light, whereas here it is called "Sohar" because it let in only a minimal amount.
Why does the Torsah insert the words "Adonei Yosef"?
Oznayim la'Torah: Potifar incarcerated Yosef without a trial - because in Egypt, a master had the right to do so to his slave.
Why did he not follow up on his anger by sentencing Yosef to death, as would have been expected?
Ramban #1: He declined to have him killed because he loved him. 1
Ramban #2: There is no logical reason for it. It was Min ha'Shamayim. 2
Seforno and Moshav Zekenim (19) #1: He saw how Yosef prospered and was a Tzadik. Perhaps his wife lied. And he only incarcerated Yosef in honor of his wife. 3
Targum Yonasan: Potifar's anger was only momentary. He immediately consulted with the priests, who concluded that what his wife had placed on her bed was the white of an egg and not semen (refer to 39:14:3:1). 4
Moshav Zekenim #2: Because it would have strengthened the rumors and shamed his wife.
Moshav Zekenim, Hadar Zekenim, Riva and Rosh (all on Bereishis 47:22, from Midrash Agadah 47:26): The priests said that if his garment was torn on the back side, it would prove that she tried to seduce him and he fled. 5
Chizkuni, citing a Midrash: Yosef was brought in front of the king. Gavriel came in the form of a man, and said that they should check their garments. If her garment was torn, this shows that he tried to rape her. If his garment was torn, this shows that she tried to force him, [and he fled].
Ohr ha'Chayim (19), Malbim (19), from Bereishis Rabah 87:9: Potifar did not believe his wife. 6
Oznayim la'Torah (on Pasuk 10): It was a miracle, as was the fact that he left prison in order to rule - in keeping with the Gemara in Kidushin, 40a 'Anyone who is tempted with Arayos, and overcomes the Yeitzer ha'Ra, merits a miracle'.
And it was to reward them for saving his life that Yosef exempted them from giving their land to Pharaoh (Refer to 47:22:1:1*). See also Ibn Ezra.
Which in fact was the case. Hadar Zekenim, citing a Midrash - Gavri'el reversed it so it was torn on the back. Riva - this is like the opinion that Yosef came to be with her, but saw his father's image and retracted. (Why was it torn in front? Was Yosef in such a rush that he could not wait to remove it? - PF) Refer to 29:20:3:7.
Bereishis Rabah: He incarcerated him only lest people question the legitimacy of his children (Matanos Kehunah - from before he became castrated. If he asserted that she tried to seduce Yosef, people will say that in the past she seduced and was Mezanah with others!)
Why did Potifar incarcerate Yosef in the prison that was reserved for the king's prisoners?
Seforno: Refer to 39:19:2:1.
Ha'Emek Davar: Because Potifar was the head executioner, this prison was under his authority.
What is the significance of the final phrase in the Pasuk "And he was there in the prison"?
Oznayim la'Torah: Yosef did not give up, but continued to serve Hashem faithfully in his new location just as he had done at home and when he worked in the house of Potifar. 1
Malbim: He was not incarcerated there. 2 Potifar appointed him to work there. The prison was of Potifar, and he needed people to work there.
Ha'Emek Davar: There are two ways if being put in the prison. (a) Some are sentenced to dwell there for a set amount of years, or permanently. They grind or do other work, and are fed in a lowly way. (b) People awaiting trial are put there, lest they flee. They do not work; they are fed from the king's treasury. Yosef was put there permanently.