"There lies a cat upon pillow mine" is actually quite perfect, albeit archaic, English word order. Actually it could still be said today poetically. I love how studying foreign languages teaches us more and more about the historical similarities between our languages!
Hmm. Well, In this case "det" translates as "there" according to the "correct" translation, and in English "There is a cat lying on my pillow," "There lies a cat, on my pillow," and "A cat is lying there on my pillow," "there" has the same meaning in each version, and the sentences differ only in rhythm, with no significant difference in meaning.
While your sentences mean more or less the same thing, the meaning of the word 'there' is not the same in each case.
In the first sentence, 'there' is used to indicate existence and is translated as 'det'.
In your last sentence, 'there' indicates position and is translated as 'der'.
In your middle sentence I think it could have either meaning and is probably only distinguishable by tone (ie, 'there lies a cat...' indicates existence; 'there lies a cat...' indicates position).