A good two hour

Mr. TomAfter a good two months, we finally found our lost key.

This one. "Good" simply means "full" here, but I think it doesn't go so well with large quantities like months. It carries the implication that you might have expected the quantity to be somewhat less than the full amount.

Mr. TomI have often seen structures like a good few year, etc.

They are often used by people with strong regional dialects and accents, as well as not much education.

The second one is correct; the adjective “good” doesn’t change the rules for “months” or “years.” I only ever seem to hear this in relation to “a good long time,” so you can see the phrase “three hours” or “two months” or “year” as a specific unit of “long time.” It’s relative in its length. You would say someone who went to the corner store to pick up a soda was gone for “a good two hours

Mr. TomAfter a good two months

Yes, you definitely need the plural here. 'good' is used this way for time, but also for distances and other amounts.

good: not less than; rather more than
(Oxford Advanced Learner's Dictionary)

We waited for a good hour.
It's a good three miles to the station.
It takes a good fou