No, that is a popular misunderstanding of it. Watching is watching, watching while walking is not being mindful, you are being mindless if you are doing that, since you are removed from the situation. Mindfulness is simply knowing that you are doing without the need to watch or reflect on the fact that you are doing.
No, it is not, it's the Buddha's definition of mindfulness, that a monk knows exactly without confusion when he is walking, knows when he is sleeping, knows when he is eating, etc. It's easy to look it up in the suttas. Saying the canonical presentation of mindfulness is somehow "Zen Buddhism" is totally incorrect.
