These are only my opinions, as they aren't explicitly explained in the film. And, I have never read the comics.
1. As Lucifer said, the family has been "Busy, busy, busy". He probably, noticed, but didn't expect Mammon to make such a bold move. Or, he didn't think Mammon could pull it off. (Not expecting the Spear of Destiny to come into play).
2. I believe that Mammon was "piercing the veil" for lack of a better phrase. He is obviously powerful, Lucifer may have known what Mammon was doing, I never figured out what God was doing at the time, as he is "all seeing, all knowing" . But as Constantine said, "God is just a kid with an ant farm." . (See above answer for my thought on Lucifer's lack of interest)
3. He didn't. He was trying to drag John to hell, and was stopped because of John's self-sacrifice to replace Isabel's soul in hell.
4. The arch-angel Michael? Maybe. Gabriel? I don't think so. But, when Gabriel tried, Lucifer said "Looks like somebody doesn't have your back anymore.". So, we will never know if it was possible had Gabriel still been in the good graces of God.
Remember, all of this is based on the context of the movie. It has nothing to do with actual any faith-based religion, or the canon of the graphic novels. And, as stated above these are only my personal opinions.
Rex
When movie hell is full, re-makes will walk the earth.
The Hellblazer universe is all about "stupid bureaucratic rules" that everybody plays. Isabel killed herself in order to not get possessed by Mammon. Which is really sacrificing herself and not committing suicide. Stupid rules...
And in the same way demons CAN go to earth but it's against the rules - Lucifer himself would punish those demons because of the "great detente" made with God.
Probably because directors or screen play writers want to be creative and add their own twist to the story. Most creative people don't want to create exact copies of a book. They want to do their own interpretation of the material.