There are actually a number of different questions there, and while I can't tackle all of them I can take a stab at a few...
In terms of getting into London, it depends which bit you want to go to. For South West London, which does contain a few of the Olympic venues, as well as a number of parks, your best bet is a local bus. The roads between Heathrow and London are pretty busy if you're not allowed to use the Olympic lanes, but for getting to the parks / gardens / etc in SW London that shouldn't be an issue.
If you want to get into the centre of London, you can either head into Paddington by train (Heathrow Express is quick but expensive, Heathrow Connect is cheaper but slower), or take the Piccadilly Line (Underground/Tube) to a variety of places. The Piccadilly line is valid with a travel card, while the train needs a supplement (Heathrow owns the spur of railway line so has it's own fares). The tube takes a little longer, but is more frequent, and may well be better for you. Get a one day travel card (either paper, or on an Oyster card, pros and cons for a single day ever visitor but that's a different question!), then hop onto the tube and visit wherever you want.
Heathrow being busy wise - the busy bit is likely to be immigration, but the day before the opening ceremony I think that everyone who could work will be, so there hopefully won't be too long queues. If you stay airside in transit, you won't have to clear immigration, just a security check. However, with 10 hours I don't think that's an issue.
Coming back, you say you're on BA, so at Terminal 5 (T5) you need to clear Conformance no later than 35 minutes minutes before your flight. Conformance is the passport/boarding card check before security, which may have a small queue (depends). Most people will be coming into Heathrow that day, not going out, so I wouldn't expect this to be a big issue, but make sure you're back from London in plenty of time! Time the tube going in, then add 50% more just to be sure, and aim to be back at T5 no less than an hour before and you ought to be fine.
With a 10 hours layover, I don't see why you couldn't have a good 6 hours in the centre of London, or 7 in SW London.
In terms of what you'll be able to see, be that at the main stadium, the tail end of the Olympic Torch Relay, or one of the many other venues around London, I'll leave someone else to answer!