I heard some noise coming out of the kitchen, so I went to have a look and noticed the kitchen door half open. or I heard some noise coming out of the kitchen, so I went to have a look and noticed that the kitchen door was half open.
"Whose" is correct there. I don't like the sentence much because it suggests an unlikely scenario where there was more than one kitchen and you went to the one with a half-open door.
Note that "who's" is a common error. The correct form is "whose," as you wrote.
I wouldn't use "whose" in either situation. If there were two doors, I'd say "I went to the kitchen on the left/right, the door of/to which ..." or "I went to the bigger/smaller kitchen, the door of/to which ...".