Bubble nests don't necessarily indicate 'happiness.' They are an attempt to attract females. So don't worry if he isn't building one. He may be too busy exploring his new home.
How often do you do water changes? A 3 gallon tank with no filter should get one 100% and one 50% water change per week.
Also, what's the water temperature? Bettas are tropical fish, and prefer temps of about 78-80F.
Moss balls are fine. A lot of people on this forum have them.
You could get a bubbler, if you want. Bettas don't need them though, since they breathe air from the surface. And if you get one, set it up so that it doesn't make lots of bubbles and push the little guy around.
As for plants, caves, etc., it depends on the individual fish. I have one male who loves caves and hiding places, so he has lots of them in his tank. I have another who ignores caves but loves plants, so his tank has lots of his favorite plants. And I have a female who is only interested in her betta hammock, so she has two.