What is the significance of the fact that Rivkah appeared on the scene before Eliezer had finished praying?
Seforno (Pasuk 45): It is a sign that Hashem agrees wholeheartedly with the request.
This is a classical case of Hashem answering a sincere Tefilah before it has been asked, as mentioned in Yeshayah, 65:24. 1 (EC)
Why does the Torah see fit to mention that Besuel, Rivkah's father, was the son of Milkah?
Hadar Zekenim (Pasuk 24), Ramban, Seforno, Ohr ha'Chayim (24), Malbim (24): To tell us that he was the son of Nachor's main wife and not of his concubine, Re'umah. 1
Malbim: It teaches that he was from Yitzchak's family both from his father (Avraham's brother) and his mother (Sarah's sister).
Oznayim la'Torah: This is why the Torah, here and in a number of other places, adds "Eishes Nachor".
Why does it say "Hinei"?
Malbim: This always denotes a Chidush. Besu'el was rich. He never sent Rivkah, rather, slaves. This was the first [and only] time she went.
Why does it say "Rivkah Yotzeis"?
Ohr ha'Chayim: She came out at this moment as a result of Eliezer's prayer.
Malbim: She left her house. 1
Also in verse 45, Eliezer says that [he saw that] she left just before he finished his prayer. He was outside the city. If the city did not have a wall, it is not astounding if he noticed her leaving her house. Why did he not say that she left the city just then, which is reasonable even if it had a wall? (PF)
