Why did Yakov inform Rachel that he was her father's brother when really, he was the son of her father's brother?
Rashi #1: He simply meant to say that he was a blood-relattion of her father. 1
Rashi #2: He was hinting to Rachel that, if her father (who bore the title 'Lavan the trickster') 2 would attempt to trick him, he was his equal in trickery 3 , whereas if he would behave towards him righteously, then he would respond accordingly.
Targum Yonasan: Refer to 29:12:1:1. He also added that Lavan would not be able to do him any harm, since the Word of Hashem was with him.
Seforno: To clarify that he had not acted incorrectly by kissing her.
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 trickster.
And why did he then add that he was Rivka's son (bearing in mind that Rachel did not know her)?
Rashi: Refer to 29:12:1:1.
Targum Yonasan: He actually instructed her to inform her father that he had come to marry one of his daughters.
Why did Rachel report to her father and not to her mother (like Rivka did, see Bereishis 24:28)?
Rashi: 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: Yakov was Lavan's uncle. He was not related to Rachel's mother, so she would not do anything for him.
Rivka 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, Rivka 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)
רש"י: אני אחיו ברמאות: לשם מה אמר לה יעקב דבר זה?
גור אריה: יעקב אמר זאת כדי שתתרצה רחל להינשא לו ולא תחשוש שלבן ירמה אותו.
רש"י: לפי שאמה מתה לא היה לה להגיד אלא לו: למה פירש כך רש"י?
גור אריה: אם לא כן- דרך הבת לספר לאמה.