WASHINGTON — Shortly before the federal government shut down at 12:01 a.m. Tuesday, a small group of top White House officials huddled in the West Wing for what turned out to be a discussion of critical importance: how to run the place. They listened as soon-to-be-furloughed junior staff members showed them how to send a news release, issue a proclamation and distribute daily reports of President Obama’s activities. One crucial piece of information was how to clear someone through the security gates for Oval Office appointments. “I didn’t know how to do that before Monday night,” said Josh Earnest, the...