QIf I put a Lucifer onto my field with Death's Breath's effect and then evolve it, does it still have Ward?

AYes, it does.

QIf I have 10 cards in my cemetery, and I put an Alucard from my cemetery onto my field with this spell, would I meet the Necrocharge (10) requirement for Alucard's Fanfare ability?

AYes. This spell would be put into the cemetery before Alucard's Fanfare ability is played, so your Necrocharge would be 10.

QI put a Mordecai the Duelist onto my field with this spell's ability. The Mordecai the Duelist is destroyed, triggering its Last Words, and I put it onto my field again. Does it still have Ward?

ANo. It leaves the field momentarily, thereby losing Ward, before it comes back onto the field.