They're the same pronoun, the "н" just gets inserted before pronouns that start with a vowel when they fall after a preposition, supposedly to aid in pronunciation (or so I've been told).
One thing to be careful about: Russian's third person possessive pronouns are indeclinable, and they are identical to the third person personal pronouns in the genitive and accusative cases (его, её, and их). However, they do not get the filler "н" after prepositions. So, for example, if you wanted to say "His father has a dog" it would be "У его отца есть собака."
supposedly to aid in pronunciation (or so I've been told).
The official version is like this: some prepositions originally had -n at the end in proto-Slavic (*sъ(n) → Rus. с/со, *къ(n) → Rus. к/ко, *vъ(n) → Rus. в/во; in this reconstruction ъ marks a very short u-like vowel). However, this n was only pronounced when the next word started with a vowel.
Eventually the forms *sъ, *kъ, *vъ came to be used before most words (because the *sъ/sъn, *kъ/kъn, *vъ/vъn alteration made no sense to the speakers), but in the most common cases n retained. Those were cases of «preposition + pronoun». So, n was reinterpreted as a part of the pronoun, and eventually this form of pronoun became used after other prepositions too (hence we have «у него», even though «у» never had n in the end in the first place).
«Него» is used after prepositions, «его» is used when it's not preceded by a preposition.
Would "Does he have THE dog" be correct as well?
Yes, Russian doesn't use articles so the same sentence would be used to mean "Does he have a dog" and "Does he have the dog". This holds true for any place you will inse3rt an article in English, but in practice context should tell you what the article is.