Why did Yakov immediately inform Rachel that he was her father's brother?
Targum Yonasan: Refer to 29:12:1:1.
Seforno: To clarify that he had not acted incorrectly by kissing her.
Refer to 29:12:2:7.
Ha'Emek Davar: Before, she merely understood that he is a relative. He now explained his exact relation.
According to Targum Yonasan, Yakov was responding to Rachel, who told him that he would not be able to stand up to Lavan, who was a swindler. He also added that Lavan would not be able to do him any harm, since the Word of Hashem was with him.
Why did Yakov add that he was Rivkah's son?
Rashi: He was hinting that if Lavan will behave towards him righteously, then he will act like the son of Rivkah, a Tzadik.
Targum Yonasan: He actually instructed her to inform her father that he had come to marry one of his daughters.
Ohr ha'Chayim #1: He told her that he has two connections to her. He is a blood relative (her father's brother), and the son of Rivkah, who was known for her virtue. Tzadikim rejoice (upon meeting similar Tzadikim).
Ohr ha'Chayim #2: Lest Rivkah say that it is lowly for Yakov to be a swindler like her father, he explained that he is Ben Rivkah
Why did Rachel report to her father and not to her mother (like Rivkah did, see Bereishis 24:28)?
Rashi (from Bereishis Rabah 70:13): Because her mother was no longer alive.
Ramban: So that her father should come out to greet Yakov and pay him respect, whereas there was no point in telling her mother. 1
Moshav Zekenim: Lavan was Yakov's uncle. Yakov was not related to Rachel's mother, so she would not do anything for him.
Rivkah went to her mother to show her mother the ornaments that Eliezer had given her, as girls tend to do. She did not know that Eliezer came from her father's family.
What is the significance of the fact that Rachel ran to tell her father?
This was the hallmark of Avraham's illustrious family - Whenever they performed a Mitzvah or a good deed, they ran (See beginning of Parshas Vayeira). When Eliezer arrived, Rivkah ran to tell her mother. By following in her mother-in-law's footsteps, Rachel proved that she was worthy of marrying her son. (EC)