Yes, easy to get around without a car, but you will likely be using some taxis since public transit hours and transit times if needing to transfer aren't all that great.

The tourist areas and main shopping and restaurant areas are safe enough by day. Even at night it is mostly ok, but don't walk deserted streets at night and be aware that some of the more interesting nightlife in the area isn't in absolutely safe surroundings.

For over two decades, I've walked from around the White House or DuPont Circle to Georgetown at well past midnight and only had one sort of close run in with someone who may have been considering using a weapon on us -- that was on a nice residential street about 50 yards from where the then US Secretary of State was living and had his own security detail outside. And that was years ago when DC was far more dangerous than it is today. Public transport in the areas you are likely to go in the day are generally fine enough from morning to closing.

I've had more close encounters with criminals with guns in nicer parts of Stockholm than I have in the parts of DC a tourist is likely to visit. And I'm the post-midnight walker doing a solo stroll way more often in DC than in Stockholm.

My most fun showing visitors around DC usually involves no car unless we are leaving the District. It would involve taxis, which are reasonably cheap compared to Swedish taxi prices.

The areas of DC you are likely to visit are far safer than Mexico City (as a whole or comparable areas in Mexico City).