1) Theorically, the only thing you absolutely NEED to buy is pencil and paper. They have online databases (and PDFs) of all the books, as well as online dice rollers. However, I STRONGLY recommend you invest in all of the equipment, as it makes the experience much more authentic.
2) Well, depends on which edition you're playing. If you're playing 1.0, you can probably get the core books for $15 each. If you're playing 3.5 ed, first off, GET USED BOOKS. You can probably get the cores (Players Handbook, Dungeon Master's Guide, and Monster Manual) for around $10-$30 each.
Dice are the most important part of the game, and you can buy them at most hobby shops for between $9-$10 a set.
Figurines are easily the most expensive part of the game, so my group improvises; we use tokens, figs from other games (like Heroscape, for example).
Don't buy the pre-printed character sheets: WotC has like seven different PDF forms that you can download and print for free.
3) You need one dungeon master and one player, and that's it. However, the game was meant to be played with five people; one dungeon master, and one of each class archetype. However, feel free to adjust the player count as you see fit; I'm currently in a party with six players and one DM.
As long as you can explain it within the story, there's no reason that people can't drop in and out. However, it is probably best if you havea solid core group of at least 2-3 players (excluding the DM. He's the only one who needs to stay to retain continuity).
4) Depends on what you use. I use 1.25 inch grid paper printed at an architectural firm (connections ftw), laminated, of course. We use dry erase markers to draw battle scenes, and that's it; it's too cumbersome to draw a whole dungeon on fair-sized grid paper. As Spike said above; your imagination is your greatest tool.
The thing you need to remember most about DnD is that the rules are less rules and more guidelines. If there's something that you don't like, don't use it. I've been a Dungeon Master for seven years now, and nothing ruins a game like having to dig through books constantly to find the one obscure roll for dodging a white dragon's breath weapon by trying to dive into your bag of holding.
*EDIT*
You also need snacks. Lot's fo them.