What is the significance of 'the fire that went out from before Hashem'?
Rashbam: It refers to the fire that Hashem sent to burn the Ketores, which 'met them' - as it were, as they entered the Heichal to burn their Ketores. 1
Moshav Zekenim 2 and Tosfos ha'Shalem (in Pasuk 2): This was Midah ke'Neged Midah .They saw fire descend from heaven, and wanted to offer a strange fire. So Hashem declared, 'Let us see whose fire will succeed - Mine or yours!'
Targum Yonasan: It is a sign of Hashem's Midas ha'Din that emerged in anger from before Hashem. 3
Why is the word "Va'tochal Osam" written with a 'Vav'?
Ba'al ha'Turim: To hint at the six sins that Nadav and Avihu transgressed ? 1. They brought a strange fire; 2. They ruled in front of their Rebbe; 3. They served after having drunk wine; 1 4. They did not (marry and) have children; 2 5. They coveted the leadership; 6. They acted independently without consulting (even each other). 3
Refer also to 10:1:1:2**.
See also Oznayim la'Torah who discusses Nadav and Avihu's sins and elaborates.
The Pasuk says that they died "Lifnei Hashem". Where was this?
Rosh, Moshav Zekenim and Da'as Zekenim (citing R. Eliezer in the Sifra): An angel struck them inside and ejected them (outside the Mishkan) before they died, to a location where Levi'im (Misha'el and Eltzafan - Pasuk 4) are permitted to enter.
Rosh, Moshav Zekenim and Da'as Zekenim (citing R. Akiva in the Sifra): They died inside (the Mishkan), and Misha'el and Eltzafan cast iron hooks in their mouths in order to take them out. 1
Moshav Zekenim: Since it was possible to remove them through Chanis (a spear? - PF), there was no Heter to enter the Mishkan.
In the Sifra, R. Akiva says that that they died inside. If so, they were Metamei the Mishkan, which is called Ohel, yet they did not cease performing the Avodah - in keeping with the command in Pasuk 7 - "u'mi'Pesach Ohel Mo'ed Lo Seitz'u Shivas Yamim"?
Tosfos ha'Shalem: They were Chayav Misah Lifnei Hashem, but they died only outside.
Moshav Zekenim (in Pasuk 4, citing the Rosh): Since it says "Al-Pi Hashem Tachanu ve'Al-Pi Hashem Yis'u", [the Mishkan] is considered (attached), and is not subject to Tum'ah. 1
I do not know why the Rosh, Da'as Zekenim and Hadar Zekenim left this difficult. Even if the Mishkan is subject to Tum'as Ohel, Kohanim may enter after the Meis has been removed, since the Ohel is not Metamei them (Gur Aryeh Bamidbar 19:14). Moreover, Korbanos Tzibur override Tum'ah, and that includes the Nesi'im's Korbanos which overrode Shabbos! (PF)
If, as R. Akiva mainrains, they died inside the Mishkan, they were Metamei the Keilim. Seeing as the Parah Adumah was not burned until the next day, how did they continue to serve?
Refer to 10:2:152:1.
Moshav Zekenim (in Pasuk 4): They were Makdish other Keilim temporarily and served with them.
Sanhedrin 52a expounds that the fire consumed "Osam", and not their garments. We already know this, since they were removed together with their tunics (Pasuk 5)?