Batman v Superman: Dawn of Justice - Why didn't Superman save his own mom?

Well, I certainly admit I wasn’t optimistic having seen the trailer, the previews, the interviews with Snyder, or the entirety of Man of Steel. But I can tell you I think I actually liked it more than I thought I would, which again, based on the evidence, was not at all. There are a few good moments. Wonder Woman is pretty much perfect. Affleck is a shockingly good Batman, although the script doesn’t do him any favors. And the movie isn’t as flat-out dumb as Man of Steel.
But...

during the battle scene, Batman is moments away from taking Superman's life when Superman utters 'Save Martha', referring to his mother whose death Lex Luthor has ordered if Superman fails to produce Batman's head to him. Batman has an instant change of heart and really does go save Martha.

But seriously, who calls their mom by their first name? And couldn't Superman just say 'Save my mother'? It would have the same thawing effect on Batman, who knows all too well how trying the loss of a parent can be.

So Superman and Batman are quick to reconcile and Batman goes to save Superman's mom while Superman goes to confront Lex Luthor.

Given that a) Superman doesn’t have to take the stairs, b) can pretty much flit in and out of places and c) has zero qualms about leaving gaping holes in buildings, I don’t understand why he didn’t just save his own mother. It would take him all of… five minutes, tops, to burst in, grab his mom, fly out while a volley of bullets bounce off his steel bod. And he could then join Batman at Lex’s new lair to show him, ‘Look, we’re a team now!’.

But Batman is sent to rescue mom, which takes a lot of bullet dodging, flame dodging and surviving an explosion. What a waste of ammo.