This Parsha was written to make known that HaKodosh Boruch Hu saved his servant Yaakov and redeemed him from a power mighter than he. Hashem sent an angel and saved him. We also learn from this Parsha that Yaakov did not rely on his own righteousness to save himself. He tried to save himself with all his abilities. Furthermore, this Parsha hints to us that everything that happened with our forefather and Aisav his brother will always happen to us with the children of Aisav. It is fitting for us to take hold of this Tzaddik's ways and prepare ourselves in the three manners that Yaakov prepared himself: prayer, appeasement through a gift, and saving ourselves by means of a war to flee and be saved.
--The Ramban

And Yaakov sent angels before him (Bereishis 32:4).
Yaakov Avinu is the root of K'nesses Yisroel (the congregation of Israel) and its symbol. With all his actions and words, Yaakov intended from the very beginning to sweeten the pains and accusations that would come to his descendants. This matter of sending messengers to Aisav and appeasing him with a gift was also beneficial for all generations to come. Whenever oppressors stand over us, the angels that Yaakov sent to his brother are awakened. They are trustworthy to fulfill their mission. They will always come forth at times of need to appease Aisav. These angels will subdue all of our accusers. That's the explanation of the posuk, And Yaakov sent angels before him. He gave them a mission for the future also: Bring the cure before the pain comes, for all generations.
--Oheiv Yisroel

I am not worthy of all this kindness and all this truth (Bereishis 32:10).
The Gemara (Sotah 5b) says that a Talmid Chochom needs to have one eighth of one eighth of the attribute of Gaava, arrogance. What is the explanation of this statement?

If you look at the 8th posuk of the 8th Parsha in the Torah (Parshas Vayishlach), you'll find this posuk, I am not worthy of all this kindness and all this truth. That is the one eighth of one eighth of arrogance which is permissible to a Talmid Chochom.

--The Vilna Gaon

And You said, "I will certainly do good with you" (Bereishis 32:13).
Hashem brings good upon Tzaddikim (the righteous) and upon Reshaim (the wicked). There is a difference, though. Hashem has good intentions when bringing good to the righteous. However, with Reshaim, the intention is not for the good itself. He wants to pay the Reshaim in this world for the few good things they may have done. Hashem pays them in this world in order to destroy them in the next world. The posuk (Devarim 7:10) says, He repays His enemy in his lifetime to make him perish. This is why our posuk uses two words for good, "Hataiv Aiteev". The good referred to here is with the intention for good and not with the intention to make him perish.
--Kedushas Laivi

Therefore, B'nei Yisroel may not eat the Gid Hanosheh (sciatic nerve)... (Bereishis 32:33).
At the root of this Mitzvah is the implication that even though we bear great suffering in our exiles at the hands of the nations and at the hands of the children of Aisav, we feel assured that we will not be destroyed. Our seed and our name will stand forever.

A Redeemer will come and deliver us from the hand of the enemy. In remembering this constantly, through this Mitzvah as a reminder, we remain steadfast in our faith and righteousness. The hint from this Mitzvah is as follows. The angel that fought with Yaakov is known to us through the oral tradition to be the Guardian Angel of Aisav. It wanted to uproot Yaakov and his descendants from the world and it was not able. The angel damaged Yaakov by touching his thigh. So too, the seed of Aisav causes distress to the seed of Yaakov. In the final analysis, however, the seed of Yaakov will be delivered from them. Similarly, we find that the sun shone for Yaakov to heal him and save him from distress. The sun of Moshiach will also shine for us and heal us from our distress and redeem us speedily in our day, Amen.

--Sefer Hachinuch

And they killed every male (Bereishis 34:25).
Why did they kill those people who were not involved with the sin against Dinah? Furthermore, why didn't they start with Sh'chem who was involved in the sin?

The sons of Yaakov had no intention of killing anyone except Sh'chem. When they set out to accomplish this, the rest of the people of the city tried to physically stop them from killing their leader. These people put themselves into the category of a "Rodaif", a pursuing murderer. The sons of Yaakov had no choice but to kill them. These people also aided Sh'chem in his kidnapping of Dinah and, as set forth in the laws of the descendants of Noach, were subject to the death penalty.

--The Ohr Hachaim HaKodosh