Excellent advice provided by @yauponredux above. At my NYC firm, we had Art History majors working alongside Econ majors (English major here myself). Any business related background necessary for the job were adequately covered in my law school corporate, tax, antitrust, securities and other business related law classes (e.g., UCC and real estate).
Other than employment at a patent law firm, your undergraduate major will not be considered by the law firm hiring you based on my experience (Wall Street law firm). For law firms I interviewed at many years ago, the biggest consideration was what law school you attended, your GPA, and journal experience. Thus, I would focus your attention on how to gain admission into the best law school you can. Back when I attended, I don't think law schools cared at all what your major was, just focusing on your GPA, LSAT score, EC's and recommendations. These days, however, I have heard that many law schools are going out of their way to recruit STEM related majors (perhaps because many of the deals that law firms are engaging in are related in some way to technology...just a guess). That being said, I would not pursue a STEM degree just for this purpose. I would pick a major which will produce a strong GPA (and those tend to be in areas that you are interested in). Good luck.
You designate ED or EA on the app so that the colleges receiving it know. There is no limitation on the timing of when you can submit your application to schools. Once the Common App opens, you can send away designating whether the application is for RD, ED and the like.