No, bettas do not really need a bubbler or air pump as they are atmosphere breathing fish - they take a minimum of oxygen from the water through gills. This is why a betta without access to the water's surface would drown.
As for live plants, sky's the limit. Bettas are not plant eaters and they don't destroy them for fun like many cichlids. In a 1-2 gallon tank I'd say keep it simple and plant one amazon sword. Don't be fooled by it when you first plant it - within a couple of months in the right light and water conditions, it'll get to be enormous. You can cut or pinch off excess growth once you're happy with the size of the plant.