Yo! I live just across the river in Arlington and work in downtown DC (well, when we're not locked down working from home). DC is a great town! I see I've duplicated some of what mwritesdragons has said, so at least we're good there
Cleveland Park would be a fine place to live. I'd live there, and I'm on the old side of the age group you're describing. It's on the metro, easily accessible to downtown (like 15 minutes from Cleveland Park to Metro Center by red line). Cleveland Park itself has a variety of condos and apartments along the main drag (Connecticut Avenue) with single-family homes just a block away. Spend some time on Google Street View for an idea. I'd live there. Now, for rent expect to pay $1500/month STARTING for a one-bedroom apartment, and over $3000/month for a two-bedroom in a newer, more modern building. Houses? Expect to pay at least $500/square foot to purchase; this puts a 2000-square foot house (largish for that neighborhood) at over $1M. Check out Realtor.com for more info. Anywhere within walking distance of a Metro station is going to cost about the same (trust me, we're looking!) with the exception of the eastern portions of the city, more about which I'll say in a bit.
The Navy Yard in Southeast DC (but still west of the Anacostia River) is recently gentrified and hip slick and cool. It's near National Stadium and currently the hot place to be. The H-Street Corridor in Northeast (approx due east of Union Station) is also recently gentrified so a lot of folks go there (Gentrification is a huge issue here, btw. All the lower-income folks are getting pushed deeper into Maryland and farther from their jobs. I don't know if that will play into your story, but it is on peoples' minds here especially in recent months). However, just a few blocks north things begin to decline. East of the Anacostia River is a very economically depressed area, synonymous around here with high crime, and I would have a difficult time believing your character lives there. There are nice parts of Anacostia (by this I include everything east of the Ancostia, such as Deanwood, Benning Road, etc., but if you use the term most locals will do a double take. There isn't much in Southwest DC (mostly federal offices), although the Waterfront has recently been updated with trendy bars, fish market, and concert venue (actually, the Anthem is an AMAZING venue). Oh, DC itself does not have many buildings over about 12 stories due to the "Height Law," which dictates that the highest occupied floor of a building cannot be more than 20 feet higher than the street it faces is wide, hence all the high-rises are in Arlington (particularly Rosslyn) or along the Red Line in Maryland in Rockville, Gaithersburg, etc.
On the Virginia side of the Potomac (have your character pronounce it "POTO-mac" if you want him laughed at for being an erudite snob) there is a lot of excitement in Crystal City and Potomac Yards area (this is where Amazon is coming in). Also, along the Orange and Silver Lines from Rosslyn to Ballston etc. is a lot of apartments/condos, with houses just a couple block off. West of there, things are mostly single-family residential until you get to Tyson's Corner, which is chock full of hi-rises, then falls to residential shortly afterward. Tyson's to downtown on the metro would take about 40 minutes. Lots of the "big money" live in Northern Virginia, and keep in mind Arlington and Fairfax counties are consistently on the list of richest counties in the US. The Columbia Pike area of Arlington is largely working class and ethnically diverse.
FYI: There are no longer any lobbying firms on K Street, although the term is still used.
I'm not a drinker or club-type, so I can't really help you with top-notch places. Check out the DCist.com, Washington Post, and Washingtonian Magazine for ideas of where is trendy. I do know that the younger crowd tends to go to H Street, U Street, DuPont Circle (although that's not as fashionable as it used to be), Upshur Street, and a lot of younger professionals live in Shaw. U Street especially tends to be too energetic for my tastes, and the older more staid folks hang out in Georgetown or Old Town Alexandria.
But just about anywhere will have happy hours, particularly along the Metro lines.
The only super A-list place I know of is The Monocle, which is next to the Senate buildings where politicians hang out. A friend of mine has a hilarious story of him, drunk off his ass, encountering Mitch McConnell and Elaine Chao (McConnell and Chao we the ones behaving themselves, just for the record).
Finally, you can get along just fine in DC without a car. It's not crazy traffic in town like New York or even Chicago, but I find some of the side roads, particularly in Virginia, confusing and frustrating beyond reckoning. I'm probably on a watch list by getting lost so many times and ending up in the Pentagon parking lot. Parking everywhere is hard to find and expensive. The Beltway traffic is as you've likely heard, although honestly I had worse interstate traffic experiences in Memphis and Atlanta. I put less than 2000 miles on my car each year, and almost never drive downtown simply because I don't have to. I tell people "I'm not crazy enough to drive here, and I'm too poor to park" but that's not really true.
Speaking of drivers, DC drivers are aggressive when it's their turn but patient when it's your turn, Maryland drivers are SUPER aggressive and BAD, and Virginia drivers are always--always--lost and confused and helpless (guilty!).
Feel free to PM me if you have specific questions you don't want to post. I'm happy to do some fact checking too, since I can be downtown on my bike in 20 minutes
ETA: Sorry, I could go on all day about this place, but I totally forgot the mention the Adams Morgan area. This area has eclipsed nearby DuPont Circle as the trendy/nightclubby area.