1. Yes, Gatwick Express is your best bet (although a bit pricey). There's a local train, too, but it takes a lot longer to get into London.
2. The London Underground and bus system are the best way to get around London, hands down. Unless you're rich, DON'T use the taxis -- they're amongst the most expensive in the world. The London Underground offers both a 1- and 3-day travelcard (and I think a 7-day card, too), which also includes busrides.
3. I always liked going to Marks & Spenser and picking up their ready-made meals.
5. I love urban vistas, but I still found the London Eye a rip-off. It's 9-11 pounds, and the ride only lasts a half-hour, with no narration or anything. If you must go, though, go at night to see London all lit up.
Must-see-and-dos in London:
-- St. Paul's Cathedral. You can climb the dome and get a great view of London. The Whispering Gallery, where you can hear someone whisper from 100 feet away, is alone worth the price.
-- Try to get a ticket to see a performance at Shakespeare's Globe Theatre. It's surprisingly cheap, but book ahead because it sells out.
-- The British Museum, which has such treasures as the Rosetta Stone and the Parthenon's Elgin Marbles.
-- The National Gallery and Trafalgar's Square
-- St. James' Park is beautiful in the summer, but I don't know how picturesque it is in the winter.
BlueWizard's blog: The Rambling Corner
HEDWIG: "The road is my home. In reflecting upon the people whom I have come upon in my travels, I cannot help but think of the people who have come upon me."