I only got my first two familiars with the planet signs in post game, for my snapdragon and one other. If you got those tickets to trade in, you can always reload and hope you get a double sign or a planet sign.
No, you can get them anytime during the game. It's all about your luck and whether a familiar with the planet sign that you encounter is willing to let you tame it.
Planet familiars get a small exp boost, and yes, generally get a bit of a advantage against other signs, but double planet/single planet familiars can cancel one another out (e.g strong against all but weak to single/double planet forgot which is which)