Clark is lying on the slab because that one Kryptonian scientist(his name escapes me) is taking some of his blood. Then Zod comes and has his talk with Kal while he's strapped up. Kal can't burst out, because aboard the Black Zero he is basically human-level.
She's fine. Her house isn't, but she is. A little shaken is all.
Supes is on his knees spitting blood because when he boards the Black Zero, he has trouble breathing and has to get accustomed to the Kryptonian atmosphere.
Agh, can't remember how it ends at the moment. Damn memory.
Faora does indeed use her dagger... well, she comes incredibly close to it. It's a good scene, you'll see.