Can't help you on bullet points one and three as I stay in hostels and have never flown from that specific airport, but I have been to London 4 times and have seen quite a bit of the city. Tons of stuff to do, all depends on what you like, if you have kids and/or older people with you and how much travelling around you want to do in your two days. I have been to (most) all of the notable museums in the area and you could easily spend 2+ days on them alone, and those are just the ones close to central London. If you're into seeing palaces and castles there are at least 10 great ones within a reasonable drive from the city. The London Eye is always fun if you've never taken a ride, especially if you're lucky enough to get a (somewhat) clear day, as it is probably the best place in the city for pictures of the Thames/Tower of London/etc.
Provide more details on what kind of stuff you like and I can probably give you better advice, and can certainly give specific advice if you're interested in the local museums.
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"Man, so long as he remains free, has no more constant and agonizing anxiety than to find, as quickly as possible, someone to worship."
Fyodor Dostoevsky - The Brothers Karamazov
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