Awakened Awareness will overwrite any previous effects that set the enchanted creature's power and toughness to specific numbers. Effects that otherwise modify the enchanted creature's power and toughness will still apply no matter when they took effect. The same is true for +1/+1 counters.

You may cast this spell with X equal to 0 if, for example, you just want an opponent's creature to have base power and toughness 1/1 but not give it counters.

The enchanted creature's toughness becomes 1 before the triggered ability that puts counters on it goes on the stack. If it has damage equal to or greater than its toughness at that point, it will be destroyed before the triggered ability that puts counters on it resolves.

If Awakened Awareness is enchanting a Vehicle that is currently a creature or it becomes a creature later, its base power and toughness will be 1/1, not the power and toughness printed on the Vehicle.