If you have pre-check and only a carryon 60 minutes is usually ample time, 75 if you're cautious. Your checked bag will remain with airline and be transfered to your next flight. You can reenter with your boarding pass for second flight.

You could Uber or, better, take Metro, which will get you right downtown as quickly and certainly more cheaply, including to the many museums on the Mall. The walk to the Metro station, which is connected to the airport, is no longer than to the taxi queue.

As for what to see, really it's up to your interests. For ease I would stick to the Mall area and nearby - there are plenty of restaurants just off Pennsylvania Ave, as well as cafes at some of the museums. They are all free to enter, though you may need timed entry tickets for the African American History Museum. You could easily spend all your time in any one of the museums, but do as you choose.