"It is my son's dog."
Translation:Det är min sons hund.
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.
Please help me clarify why "sons" is "söner" and "son's" is "sons"