"The house next door is correct. It means that particular house.

Both are correct, but "the" is more likely. There aren't a lot of houses next door, so you probably know which one. The sentences are fine the way you wrote them. You don't have to put "last week" at the beginning.

I see, thank you!

I think both are acceptable, though it should be rephrased slightly different: Last week, John bought the house next door. Last week, John bought a house next door. Using "a" is slightly more correct, but "the" would also work depending on the conversation.