Firstly, is the hound even a creature? If so, does it get advantage due to being invisible, and opportunity attacks — seeing as how animated objects get opportunity attacks?
It is probably not a creature.
The term "creature" is not explicitly defined in 5e, so we have to look to other spells and situations for analogous situations.
Spells such as Unseen Servant and Bigby's Hand create entities that have some of the properties of creatures, like HP, AC, and grappling, but they are not creatures. Likewise, while the hound can make attacks, it doesn't have other game statistics like ability scores.
On the other hand, the spell Simulacrum explicitly states that its creation is a creature:
The duplicate is a creature, partially real and formed from ice or snow, and it can take actions and otherwise be affected as a normal creature.
Mordenkainen's faithful hound does not have such text, and neither do Unseen Servant or Bigby's hand.
It does not get opportunity attacks, and advantage is probably a DM ruling.
Given that it's not a creature, the rules for opportunity attacks don't apply. Note that animated objects (from the Animate Objects spell) don't get opportunity attacks either, as you can only command them to attack on your own turn.
RAW, the invisibility rules require the attacker to be a creature, and since the hound is not, it doesn't get advantage. I think that is open for DM interpretation, however.