FROM HIS OFFICE ATOP A BURBANK high-rise, Adam Fogelson has a bird’s-eye view of three major Hollywood studio lots: Warner Bros., Disney and, perhaps fittingly, Universal, where he worked for 15 years, including four as chairman, until he abruptly was ousted in September 2013. Now, Fogelson, 48, is trying with STX Entertainment to do what Relativity Media and even the old DreamWorks couldn’t: challenge the traditional studios by moving in on an area they largely have abandoned, midbudget ($25 million to $75 million), star-driven dramas rather than superhero pics and prebranded tentpoles. And he’s making a lot of them. The 65-employee STX, founded in spring 2014 by industry veteran Robert Simonds and backed by, among others, TPG Capital and China’s Hony Capital (with significant film financing coming from China’s Huayi…