Nobody starts on the 0 space. The only time you can be on the 0 space is during the end game scoring when your cube can be pushed down to the 0 space - see the scoring example on p5 of teh rulebook.
From the rulebook p6 Propraetor: "Important: ... At the start of the game, no player has a marker on the board, you do not start from the 0 space. This is important, as the markers on the 0 space are worth points at the end, while those out of the game are worthless."
So in your examples:
1) If you are on the 0 space then by definition it is the end of the game and you are scoring therefore you have no opportunity to play cards.
2) You have no influence in an area. You tap one card to raise area influence by 1. Your cube moves from your pool of cubes and is placed on the 1 space.
3) You have no influence in an area. You tap three cards to raise area influence by 3. Your cube moves from your pool of cubes and is placed on the 3 space.
I could have sworn I read a post from the designer saying otherwise. But now I can't find it or maybe I misunderstood what he was saying. In any event, all this makes perfect sense to me. Thanks again Damon.