When Abraham's servant meets Rivka's family, her father Besuel appears as an actor only once , and in the middle of the story, but not afterwards. Rashi solves this problem by introducing a midrash: But, where was Besueil [her father]? He wanted to prevent [Rivkah's leaving] [therefore] an angel came and killed him. 
However, this doesn't explain why no mention of Besuel is made at the beginning of the story. For example: After Rivka gives water to the servant and his camels she runs to her mother's house; afterwards Lavan alone comes out to greet him 
Indeed, Robert Alter says there is good evidence that his appearance in the middle of the story is a scribal insertion (ie: in the original text, he wasn't mentioned AT ALL for reasons given below.) Evidence of this includes that previously mentioned fact that Rivka's home is called "her mother's house"
and that when Besuel does get his mention he shares the scene with Lavan , yet the introductory verb stays singular (וַיַּעַן לָבָן וּבְתוּאֵל וַיֹּאמְרוּ)
Relying on ANE documents, Alter says that it was the common cultural practice for the eldest brother to lead bridal negotiation, while the father sat silently. He suggests that a scribe, writing after the custom had disappeared, corrected the text by inserting a mention of Mister B. 
[Note: This is all speculative, and I have no idea if Alter is right or wrong. Also, I don't care. The Torah is still the torah even if corruptions and maculations were introduced along the way.]
1 - Verse 50: "Lavan and Besuel answered and said..."
2- Other Rishonim say the opposite, and claim that Besuel name is omitted because unlike the mother and the brother, he supported the match. You are invited to contact your second grade rebbe and ask why these Rishonim are not taught. I'd love to know.
3 - Yes, thanks, I know that Rashi has explanations for Besuels absence in both cases, that do not rely on Alter's spin.
4 - He also suggests that Besuel had died before the servant arrived. This, and not the ANE cultural practice, is a better explanation for Rivka running to her mother's house, and a fatherless Rivka is nicely symmetrical to the motherless Issac.