The kid is their Lex Luthor. Technically I think it's supposed to be Alexander Luthor, the Lex everyone know's son. I know, kind of stupid.
Is that officially supposed to be Doomsday? From the scenes leading up to his appearance, it looked like Lex was reviving one of the Kryptonians, or maybe Zod himself. Kept in cold storage by the government most likely.
I still don't like that they are billing this as a dark and gritty world, and a punch up between Superman and Batman until a bigger threat shows up. I just don't buy it, it makes both characters to myopic. "He doesn't act like me, so he must be eviiiiil despite all actual evidence to the contrary.'. Bats and Supes were always two sides of the mirror of justice. Their methods and attitudes, as well as the enemies they fought may be different. but they share so much. Two people orphaned at a young age who had the power and ability to make their world better, and felt honor bound to do so, even... and especially when doing the right thing is the hardest thing you can do, and some people will hate you for it. They may be very different, but they understand each other, and have enough respect for each other to not let their differences turn them against each other.