The correct answers are The old castle, which was built in 1492, is supposedly haunted. and The ghost that haunts the west wing hates loud noises.

Which is correctly used in the first sentence to add extra, non-essential information (a non-defining relative clause).

That is correctly used in the second sentence for a defining relative clause about the ghost.

The third sentence is incorrect because you cannot use that in a non-defining relative clause (after a comma). The fourth is incorrect because which should not be used for people (the tour guide).