1. I'd say his style is "matter-of-fact about the remarkable." Oftentimes his stories deal with either real-world implications of a fantasy idea, or having a "normal" person interact with the fantastic, accepting it and dealing with it (usually) without going "Oooooooh! But that's... that's impossible!"

One of his short stories that comes to mind is about an American going on a walking tour of England. There are lots of jokes about boring towns and English weather and English food, and then he drops into a pub and chats with two quintessentially English bar guys who drink and dither about the working life of a Cthulhu cultist. ( "Shoggoth's Old Peculiar" )

Or, to flip it, The Sandman, the graphic novels he wrote a while back, have a protagonist who is basically the God of Dreams, but a lot of the stories are about his interacting with mortals.

1. Magic in the everyday world around us. Not magic in the sense of Harry Potter, or even magical realism like Borges. More like an everyday acceptance of the strange. He once wrote a book called "The Graveyard Book" (ostensibly for children but still quite good). It's basically "The Jungle Book," but instead the child is raised by ghosts and vampires and the like.

2. Generally just coming up with interesting twists on mythology. He is definitely NOT a fantasy writer like, say, Brandon Sanderson or Jim Butcher. Barring Sandman, he doesn't really write serieses. He is utterly unconcerned with building coherent magic systems, or fantasy, perpetually-medieval worlds. Most, if not all, of his stories have some sort of grounding in the real world.

A good example of this is "Good Omens," the book he co-wrote with Terry Pratchett. It's a story about a Cold War between demons and angels, a pair of "agents" who become friends, and an Antichrist that is misplaced and raised in a rural, bucolic town.

1. Hm. It depends on what you like.

If you like Terry Pratchett, then I'd say start with Good Omens- it's an excellent collaboration between the two.

If you want something epic and dark and don't mind a bit of rambling, start with American Gods.

(Concept: All the gods that various immigrants have ever worshiped still live and exist, but as they're mostly out of fashion they hang out in isolated niches in middle America. The protagonist meets a man calling himself Mr. Wednesday, and they go meet various gods to prepare for a confrontation with New Gods such as the gods of Media and the Internet)

Otherwise, maybe one of his books of short stories- Fragile Things is pretty good.

One note: I suggest AGAINST starting with Neverwhere. Many fans love it, and it's definitely a kind of cool book, but in my opinion it's not very well constructed. It's the novelization of a TV series he worked on, and you can definitely see the invisible lines where one episode ends and another begins.