In the Buffyverse, blood is life, so do you actually need blood to close something? It's not bringing something to life, it's bringing something "dead." Angel/Angelus's blood was used to open Acathla's mouth and Dawn's or Buffy's blood was used to open the Hellmouth. Buffy said that blood between her and Dawn were the same, so she could close the portal in "The Gift," but she doesn't seem to be bleeding... Maybe the portal needed Buffy's life energy only and life energy comes from blood. In Angel's case, I think Acathla needed the body of the person or thing that opened it... complete with his blood. I don't think he can just cut his hand and let it bleed on Acathla. Acathla needs his body.
About the killing part, Wesley said in "To Shanshu in L.A." that something that's already dead can't actually "die," so technically, if Angel had no "life" left in him if Buffy had killed him, he wouldn't actually be "dead."