What exactly did Pinchas see?
Rashi: Pinchas saw what was happening and remembered the Halachah. He said to Moshe, 'I learned from you that "ha'Bo'el Aramis, Kana'im Pog'in bo" (If someone has relations with a Nochris, any zealot is permitted to kill him). 1 So (when he saw that everyone else remained silent, even the lions of Yehudah - Targum Yonasan) he took his spear ...
Under certain circumstances - See Sifsei Chachamim.
What is the difference between "El ha'Kubah" and "ve'el Kavasah"?
Rashi #1: The former means that he followed them 'into the tent', and the latter that he pierced the two adulterers through their stomachs 1 (through their genitals).
Rashi #2: The fact that the Zero'a the Lechayayim and the KeivahHe was rewarded for this by the Keivah (the stomach) from all Chulin animals are included in the Matnos Kehunah is a reward for this act of Pinchas. 2
What are the connotations of "Vayakam mi'Toch ha'Eidah"?
Rashi (in Sanhedrin, 82b): It means that he arose whilst the Beis-Din were judging the sinners.
Why does the Torah need to teach us that Zimri pierced them through their genitals?
Rashi: To make it clear to one and all that he did not kill them for any ulterior motive, but because of the sin that they were performing.
Rashi writes that Pinchas told Moshe, I learned from you, 'ha'Bo'el Aramis...' One who rules in front of his Rebbi is Chayav Misah (Eruvin 63a)!
Moshav Zekenim: When there is Chilul Hashem, we do not show honor to the Rebbi (i.e. we immediately stop the Aveirah).