Can you help with my questions about travelling in London, UK?

My friend suggested I ask the collective here. I apologize if I’m posting in the wrong place.

My husband and I are travelling to London in May, 2014 and were planning to stay with family, but unforeseen circumstances have eliminated that option and so we are starting fresh and completely at a loss.

1) Is there an area of the city you’d recommend as a base for touring?
2) We prefer inexpensive but personal lodgings. Do you know of a nice guesthouse or pub accommodations in that area? (Not interested in hostels)
3) What is the best way to get around London? I’ve heard something about transportation passes. We don’t want to rent a car. Where would I purchase something like that?
4) What are the “don’t miss” places to see? Not necessarily the normal tourist destinations, but places you would take a guest if you are a local or places you’d revisit if you have been a tourist.

We’re a bit anxious about plans having fallen through for our first trip to the UK but excited about the adventure opportunity.
Any and all help is greatly appreciated.