Does anyone know how Killing or Sparing Capt. Slate affects the outcome of the game ?
The outcome of the game (ending) is not affected by this decision.
When you reach the jail in Fink Mfg. if you did not kill Slate you will see him in a jail cell. He appears to have been lobotomized as he does not acknowledge you in any way and there are a variety of medical tools nearby, Elizabeth will mention that you were right, sparing him was no mercy. You are given the non-obvious choice of shooting him at this point, Elizabeth will mention that it was what he had wanted. Either way it does not change the overall storyline. Interestingly, however, killing him then, or after entering the tear, will kill him in both universes.