What exactly, is "Chelev", as opposed to Shuman"?
Ramban (9), Moshav Zekenim (9): "Chelev" is fatty membranes that peel from the animal's flesh easily, whereas 'Shuman' is the fat that is closely attached to the flesh. 1
What does "v'Es Kol ha'Chelev" come to include?
Rashi #2 (citing R. Akiva in Toras Kohanim): It comes to include the Chelev on the intestines exclusively. Refer to .3:4:2:1*.
Moshav Zekenim (citing R. Yishmael, Chulin 49b): Just like the Chelev covering the Kerev is a membrane that can be peeled off, also all such Chelev (of a Shelamim is offered, and is forbidden to eat even in Chulin). R. Akiva includes only membranes spread over limbs, that can be peeled off. Toras Kohanim switches the opinions..
And certainly the Chelev on the intestines (Sifsei Chachamim).
Why does the Torah command to burn the Chelev on the Mizbe'ach?
Moshav Zekenim #1: It comes to atone for sin that one does amidst obesity - "va'Yishman Yeshurun va'Yiv'at" (Devarim 32:15). Therefore, one offers Chelev, and it is as if he offered his own Chelev.
Moshav Zekenim #2: Chelev on the Kerev, which is full of excrement, should come and atone for the sin of a man full of reeking odors. The two kidneys, which counsel, should come and atone for kidneys that counsel sin and deceit. Yoseres ha'Kaved should come and atone for hardening one's heart and ear.
רש"י: אסורה מכאן להתקרב שם: מנין לפרש כך?
גור אריה: משמעות הפסוק-'אסורה ולכן אראה', וקשה שאפשר לסור גם לצד אחר ולא לראות? אלא 'אסור מכאן להתקרב לשם'.