1. DC functions pretty similar to NYC in that regard, there are neighborhoods that are very popular with new grads. Also while DC doesnt have many finance jobs there are a ton of consultants and lawyers. Similar to NYC it isnt uncommon for new grads to live in an apartment that has been converted to add an additional bedroom (or has a sunroom/den used as a bedroom), generally you get a bit more space than you do in NYC but in my experience prices arent wildly different you just get a bit more. Unlike NYC the areas outside of DC tend to be nicer (e.g., Arlington, Bethesda, Alexandria) but the key driver of prices in DC is metro access and which line you are on (red, orange, blue > green). Your uber story is also the exact same with DC, pre uber getting a cab back to VA/MD was a pain in the ass, now it might take me only a few minutes longer to get to a bar in DC as some of my friends who live in the city.
2. We obviously have the height restrictions but on the whole it seems DC has been relatively developer friendly, the big drivers in recent years have been Navy Yard and the Wharf. People have also made a ton of money buying your typical row home, renovating it and turning it into condos. VA/MD have also seen quite a bit of development on the metro lines, the silver line being the most obvious example.
3. The wharf seems popular but is too commercial for my taste. Not sure either are really considered cool? they are both a bit out of the way from the "cool" areas but Navy Yard in particular seems to be changing as places like Dacha pop up but I only find myself out there after a baseball/soccer game.
4) Logan circle is very popular with young people that have money, the U street corridor is also very popular. At least in my social circle the nicer parts of north east have been popular for starter homes, or moving to Virginia (we + several of our friends landed in alexandria). Once you are looking at putting your kids into school I think the general consensus is you are looking at outside of DC or private school (although iv heard some of DCs schools are getting better?).
5) All the Virginia schools are big feeders, anecdotally UVA grads tend to live in the city with VT/JMU grads landing in arlington. (VT/JMU also seem to have a larger % of grads end up here vs UVA which tends to see people disperse a bit wider). GMU, GWU, and Georgetown are also well represented. But you also get a ton of folks from all over the country coming here to go into politics/non profits/law etc.
6) North of colombia heights and northeast can be a bit sketchy but in general the more north and/or east you go the worse it is and anything on the other side of the anacostia river is a no go. Gentrification is changing it a bit but DC is a pretty typical railroad tracks city, west of union stations is nicer than east. Obviously these are generalizations but on the whole iv never had any issues (but also a privileged white male)