The cat's very ill, poor old thing.

No.

Johnson13should there be the indefinite article ("...a very ill...")?

That wouldn’t be very natural. You need The cat’s very ill. Poor old thing!

Johnson13In this informal usage, THING obviously refers to CAT, one single cat; should there be the indefinite article A VERY ILL...?

No, because 'poor old thing' is not a continuation of 'very ill'. It's a separate comment.

You may be thinking that the sentence is about a very ill, poor, old thing, i.e., a thing that is very ill, is poor, and is old

Poor old thing!
But that's not a complete sentence. I would go with the sentence below.

'The cat's very ill, poor old thing!'

"poor old thing" expands on the first part of the sentence.

canadian45But that's not a complete sentence.

Such interjections are traditionally used in isolation, and rightly so.

CalifJimNo, because 'poor old thing' is not a continuation of 'very ill'. It's a separate comment.

But why does it have to be a separate comment? Why does it have to be an interjection?
I think the original sentence is fine as is. That's not to say that an alternative can't be offered.
But I don't agree that the original sentence is wrong and has to

Thanks.

Does it mean THING does not refer to CAT?

Johnson13Does it mean THING does not refer to CAT?

"thing" does refer to "cat".

The sentence is from the OALD, and it says it's countable; do you think it's a typo?