As with most adverbs, 'completely' goes before the main part of the verb which it is modifying. He completely forgot her birthday. He had completely forgotten her birthday.

Yes, I agree with Marcus. The first sounds good, the second does not. You could, if you want, put 'completely' in a completely different place... I forgot her birthday completely! We often say this. (Well, I do! I'm always forgetting birthdays!)

The first sentence sounds much more natural. That's a very common expression, good question!