What is the significance of "in the evening she comes, and in the morning she returns"?
Torah Temimah citing Megilah 13a: Amidst the detriment of the Rasha we hear his praise, that he would not have relations during the day.
Malbim: Even though in the morning she will go to Beis ha'Pilagshim and be stuck there [permanently], they went willingly 1 .
Malbim (13): She went willingly due to the gifts; they gave to her whatever she requested.
What is "Beis ha'Nashim Sheni"?
Rashi: It is Beis ha'Nashim ha'Sheni. (The first was for girls who did not yet go to the king.)
Why was she sent to "Shomer ha'Pilagshim"?
Vilna Gaon: One who had Bi'ah with the king is forbidden to all [except a king], like it says 'Avishag 1 was permitted to Shlomo, for a king may use another king's staff. She was forbidden to Adoniyahu, for he was not a king' (Sanhedrin 22a). Therefore, she was not sent home; she became a concubine.
"Veha'Melech Lo Yeda'ah" (Melachim I, 1:4)! There is a Kal v'Chomer. Avishag was forbidden to commoners, even though David never had Bi'ah with her, and all the more so one who had Bi'ah with the king! (PF)
Why does it say "Lo Savo? Ki Im? v'Nikre'ah v'Shem"?
Vilna Gaon: There was a fixed time for Besulos to come - after 12 months. A Pilegesh came only if he called for her; their names were written by the king.