Fort Worth has several great neighborhoods, but for young professionals, I would say West 7th Street area is the place to be. There are great apartments and condos in the area with lots of restaurant choices and entertainment venues. You can walk to most everything. Lots of great shops, wine bars, spas,etc.
Depends on your definition of cool. I'd say if you are single and have the cash. Any of the downtown condos or the condos in the West 7th area would be pretty cool. Other than that, anywhere in the cultural district.