Subordination in Chinese
If I extrapolate what you are doing, it actually becomes "unnatural" to me, because the subordinate clause is inserted in front of the noun that is being modified, but after the original verb. Your examples can be translated like the following:

I know a person.
我認識某個人

I know a person who has a dog.
我認識某位有養狗的人

I know a person who has a dog which barks at a cat.
我認識某位養有會對貓吠的狗的人

I know a person who has a dog which barks at a cat which lives in a house.
我認識某位養有會對那隻住在屋裡的貓吠的狗的人

I briefly add/tweak some charaters so that it makes more sense to me.... I'll leave it up to you to see if you can just "get the idea". :)

If I "naturally" say these, it'll probably not be all subornate clause but shorter SVO phrases:

I know a person
我認識某個人

I know a person who has a dog
我認識某個人，有養狗

I know a person who has a dog which barks at a cat
我認識某個人，養的狗會對貓吠

I know a person who has a dog which barks at a cat which lives in a house
我認識某個人，那人養的狗會對貓吠，那貓還住在房子裡