O_O.
1. The most central is London City, but I've never been there always gone from Heathrow or Gatwick.
2. If you want cheap accomodation then it's going to hard for you because London isn't cheap!!!! I just did a google search and got lots of results, so...just google "cheap london hotels" or "cheap london bnb's"!
3. I don't see why not
4. For me.....I would be 30% about being able to make it on that day, but perhaps I can