The Stars on Pokemon Rumble World (for levels) are represented by rarity.
For example, stages might have 1, 2, or 3 stars. The more stars, the rarer Pokemon to find - with 3 stars stages to represent legendaries for boss stages.
On moves, it represents attack power.
If they're blue, it represents STAB (Same type attack bonus). This means, if the Pokemon has a move as the same type as they are, if will increase power (such as kingler with water pulse).
If they're white or blank, they're other moves that they've learned.
Fevers happen at random times and they occur more often as you continue in the game. Same with regular stars.
Source: http://www.gamefaqs.com/boards/141816-pokemon-rumble-world/71596764
Hope I helped!
