American authorities sent David C. Headley, a small-time drug dealer and sometime informant, to work for them in Pakistan months after the Sept. 11, 2001, attacks, despite a warning that he sympathized with radical Islamic groups, according to court records and interviews. Not long after Mr. Headley arrived there, he began training with terrorists, eventually playing a key role in the 2008 attacks that left 164 people dead in Mumbai.