Not a chip off the ol' block

1) Rashi writes that Yaakov davened that his name not be connected to  the yichus of Korach. What difference does not mentioning his name make -- we all know  Korach came from Yaakov?  Maharal in Gur Aryeh (discussed back in 2009) explains that with great yichus comes great responsibility -- you have to live up to the privilege you have been afforded.  Had any Ploni started a rebellion, it would be bad, but when you are "ben Kehas ben Levi" and start a rebellion, it's even worse.  Yaakov did not want to be associated with condemning any Jew, even Korach.  Therefore, he asked that his name be omitted from the chain of yichus that served to highlight the severity of his sin.

R Baruch Sorotzkin has a different hesber.  Eisav was born first and should have been the bechor charged with the responsibility for avodah. Yaakov realized this would have been a disaster and therefore, when the opportunity presented itself, he demanded  that Eisav sell his birthright.  Under the guidance of Rivka, Yaakov later went ahead and took the brachos that Eisav might have gotten.  Korach might have been thinking to himself that he was a chip off the old block of great grandpa Yaakov -- he too would seize the privilege of avodah from his relatives and restore it to himself, to the person most worthy.  Yaakov davened that the kochos he used in his struggle against Eisav not be taken and perverted and used improperly by Korach.

2) Last post we discussed Ramban's view that even though there is an issur of offering ketores outside its proper place, and even though Moshe was not commanded by G-d to do so, Moshe took the initiative of suggesting it as a test to prove that Korach and co. were wrong. It could be that Moshe did so as a horaas shaah, similar to Eliyahu offering a korban outside the Mikdash in order to prove the prophets of Baal false.  The gemara (Sanhedrin 89) writes that you should obey a Navi who tells you to do something wrong as a horaas shaah only if that person is muchzak as a true Navi.  The very fact that the 250 people accepted Moshe's horaas shaah suggestion in effect validated him as a Navi and undermined Korach's argument (Chasam Sofer).

3) Why was the punishment of Korach's followers that they descend "chaim she'ola," to gehenom while still alive?  See Maharal in Tif Yisrael ch 18.  The second Belzer Rebbe suggested that Moshe, like Yaakov, as we discussed above, did not want any Jew to suffer at his behest.  So long as a person has life in him, there is the possibility of teshuvah.  Therefore, Moshe asked that the rebellious plotters remain alive even as they had to go to gehenom, so as to keep the possibility of repentance open.