"It is my son's dog."
Translation:Det är min sons hund.
Det does not refer to the dog here, it is a formal subject much like it in It is raining.
But wouldn't Den be acceptable as well if, say, the context were about this specific dog?
If the dog had already been mentioned, yes. If not then 'den hund är...' might sound like 'the dog is my son's dog' because 'den' has to refer to something specific.
We can't say 'den hund är …'.
It's possible to use 'den hund' for 'the dog' if a relative clause follows: den hund som sitter där … 'the dog that is sitting there'
It's also possible to use den + determinate noun, but that will mean that: den hunden är gammal means 'that dog is old'
What you probably meant to say is that if the dog had been previously mentioned, it could have worked, like this: Hunden är gammal, men den är fin 'The dog is old but it's nice'.
It's not possible in a standalone sentence like this one though.