How did they kill the horse?

How was Khartoum killed? And how was his head placed in Woltz's bed?

Killing a horse is not that hard, especially if the horse is trusting. Removing its head rather noiselessly is much more difficult. Sneaking a horse head into Woltz's bed with him in it is very difficult as well. Doing all of that in a time period of after dinner to dawn makes it all the more challenging.

Thinking about it, some, if not all, of the staff must have been bribed and/or strongarmed into looking the other way at the very least. But how could that be put together on such short notice? Did Tom send someone to grease them in between meetings? It doesn't seem feasible, but I see no other way.

And I'm still perplexed by the removal of the head. The stable was not that far from the house, so a chainsaw or similar device would have been heard. Using axes would have taken far too long.

He was fed Connie's cooking.

The novel hints that it was an inside job, which is what made Woltz come around. Nothing in the novel or the film says the horse was killed the same night that Tom had dinner with Woltz.

I guess it's not overtly made clear, but I have always been under the impression it was the same night.

How could it be the same night? Vito did not know of Tom's meeting results until he returned to NY and told Vito about those results and Vito formulated a plan.