1) We really liked where we stayed (The Strand Palace Hotel--east of Trafalgar square, centrally located near Temple and Covent Garden), but I don't know of anything off the top of my head that would make any area that I know of better or worse for staying w/ a child. Sorry.
2) Our hotel (Strand Palace Hotel) was within walking distance (2 minutes tops) of Covent Garden Station, which is on the Piccadilly line (runs from Heathrow, so very convenient). We were also less than a 5 min walk from Temple Station, which has the District and Circle lines, which run to Westminster one way and to Tower Hill (Tower of London, Tower Bridge) the other way. Also, it very walkable from that area to Westminster, Piccadilly, St. Paul's, Millennium Pier (where you can take a boat to Greenwich).
Hope this helped--we were VERY please w/ our location and hotel.
