Their size is unconfirmed although the engine supports infinite size. Whether or not this will become a feature is unknown.
Every planet in the game has a threat level, from 1 to 100. Level 100 threat planets are intended to be difficult even for players that have end game gear.
It is confirmed that you will be able to rename your home world. Whether or not you can rename any planet is unknown.
The current confirmed biomes on planets are: Forest, Arctic, Tentacle, and Desert.