Is it possible to explore London during a 10-hour layover at Heathrow?

This is absolutely doable. Ideally, check the bags through. Even if you have to get your bags, there are left luggage facility in each terminal, where you can leave your bags for the day for a few quid (under $10) each.

There are many ways to get from Heathrow into London - depending where in London you need to get to. Search this site - there are plenty of questions about it. Plan on returning to the airport about 2.5 hours before your flight if you need to get your bags and check them in again - or about 1.5 hours if the bags are checked in through and you have boarding passes.

I have done this with a much shorter connection - only had about 6 hours connection with bags checked in through and boarding pass in my hands for the second leg. It took me about an hours to get through the immigration/passport control - no need to wait for bags - and out of the airport. Heathrow Express got me to Paddington about 20 minutes later - roughly about 1.5 hours from the time the flight landed to the time I got off the train at Paddington. I had about 3.5 hours in London then before getting a train back from Paddington to Heathrow - 15 minutes journey time plus a bit of walking on the other end. Took me about 30 minutes through the security in Heathrow - and I was at the gate about an hour before the departure - 15 minutes before the boarding started.