If I recall correctly, the dog has to fit in a carrier under the seat, there is a fee, and you need to call ahead to make sure the dimensions of the carrier for that particular flight and to make sure that there are no more than 2 pets already in 1st for said flight... Also yes it counts as one of your carry-on pieces.
As a side note, some FAs seem totally fine with non ESAs sitting in laps, but others will throw a fit. I once saw an adorable chihuahua nearly bite the hand off a fellow FC passenger and then an FA who got too close to its owner.