Sounds like the Midtown, Montrose, and Galleria areas would be the best for you.
The Galleria area and right along Washington Avenue have the advantage of being close to Memorial Park while the Montrose and Midtown areas have the advantage of being close to Hermann Park. Midtown & Montrose would also be closer to the main rail line as well as downtown.
For nightlife and things outside of your door, Montrose, Midtown, and Galleria areas are probably the best shots. I'm more of a homebody, so please take my advice with a grain of salt. The area along Washington has nightlife as well.
Graphic Designer positions will be spread out all over town, but the heaviest concentrations probably will be center and west.
There are a fair number of apartments, condos, and townhouses right in the Galleria area. As long as you aren't in the residential home areas, there shouldn't be too many kids.
As long as you aren't on the North, far West, or Southeast side of town, access to beach access will be about the same (an hour) for most areas.