HELP! Need transit directions in/around London

[Anne]

Actually I need these for a fan fic I'm working on -- though I have 
visited London, we stayed in a different part of town (Pimlico) than 
where I'm interested.

I know that in PS/SS, Harry and Hagrid stopped for a snack at 
Paddington Station before Harry took a train from London back to 
Little Whinging. But I want to get Harry INTO London on a train, from 
Little Whinging. So, I have several questions:

1) What train line(s) (Britrail?) would go to Paddington Station from 
anywhere in Surrey?

2) Do any train lines from Surrey go to other stations, and which 
ones?

3) How would a person get from Paddington Station to Diagon Alley? 
Would one take the Underground from Paddington to Charing Cross? (I 
am assuming Diagon Alley is not distant from Charing Cross station, 
because The Leaky Cauldron is on Charing Cross Rd.) If so, which 
underground route? If not, would one take a bus? Would it have to be 
a double-decker bus (open-topped) to accommodate Hagrid??

I don't mean to be a pest; hope someone can help. Thanks in advance. 
Feel free to email me off the list.

Anne U