re: Why would the president fight so hard for contraception mandate?

They're framing a narrative. Young single women (incl single mothers) grew up with all kinds ideas regarding what they're entitled to RE: sexual health and reproduction. It resonates, and women 18-33 or so are growing into an important voting block. They have more money, independence, and political conviction than at any time in history.

"War on women" rhetoric may well end up paying better dividends than racial identity/grievance politics ever did for either party. And you'll always have some old fn relic say something during an election to fan the flames (i.e. real rape vs. other kinds, etc).