I have written about the illusion of a state of Happiness in the past here. In doing so, I have always pointed out that it's a term without a meaningful definition. However, it has always seemed to me that happiness implies, for some, an imaginary infantile state in which all wants and needs are met rather than the state of stress, difficulty, and challenge which many of us seek.
For a simple example, I have a friend who never feels "happier" than when struggling for hours in a fruitless effort to master a Chopin piano piece.
Transient joys and delights certainly occur, as can periods of contentment with their implication of acceptance of, or resignation to, the limits of reality.
Leszek Kołakowski poses the question Is God Happy? (h/t Althouse) as a way to reflecting about the human capacity for happiness. He says:
Happiness is something we can imagine but not experience. If we imagine that hell and purgatory are no longer in operation and that all human beings, every single one without exception, have been saved by God and are now enjoying celestial bliss, lacking nothing, perfectly satisfied, without pain or death, then we can imagine that their happiness is real and that the sorrows and suffering of the past have been forgotten. Such a condition can be imagined, but it has never been seen. It has never been seen.
Kolakowski clearly adopts the definition of happiness to which I alluded earlier, ie, the serene absence of any disturbing thoughts or feelings. Sounds more like a mindless beach vacation to me than something anybody would aspire to for more than a few days. Sounds like heroin.
Regarding the question re God's happiness, it's an absurd question. God is not human.