I don't have the answer to your question, but I don't see the connection between mindfulness and Zen to be as direct as you're suggesting. I've been highly influenced by a lot of the mindfulness writing out there, but Zen actively turned me off for a long time, because it's a lot of "deep" paradoxical puzzle/koan things, anecdotes that make no sense, emphasis on "enlightment" etc. Contrast that with a lot of modern day mindfulness stuff which is highly, highly influenced by Buddhism but basically extracts the most practical stuff, meditation practice, awareness of thoughts/feelings, recognition of egotistical cravings, acceptance of impermanence, etc.
That depends a lot on the Zen school and the texts you were reading. Zen is based on direct experience. That being said, in some schools, such as Renzai, koans are used as a tool. However, most sangas you attend, especially in the west, will focus more on the direct experience of meditation to start, which is more or less the same sort of thing one is taught by most mindfulness / MBSR / MBCT regimens.
Bear in mind that Zen is a form of Buddhism, and as such, has religious and philosophical depth. That being said, as with many such traditions, the point is for a practitioner to directly experience the religious revelations if the practitioner is drawn to the religion. Most sanghas are perfectly happy to teach a secular lay practitioner how to meditate if he/she is open, honest, and respectful regarding intentions. In that case, you won't be given koans (assuming the school of Zen you are visiting uses them) or dharma talks. You'll be taught how to meditate. Just like with mindfulness training, that starts with the counting and experience of the breaths.
