Here are some answers (or attempted answers) to the four questions you pose. God knows best.
1. You ask how to “fit” your daily prayers into an already tightly packed schedule. One of the Catholic saints—Francis de Sales, perhaps—said that everybody should pray at least a half hour a day, except for people who are really busy, and they should pray an hour a day! This will seem to you unrealistic, I’m sure—until you begin to see more clearly where Reality lies. In the meantime, you can work toward this ideal incrementally. I would start by trying to set aside ten minutes of quiet, concentrated time for the Jesus Prayer the first thing in the morning, and then again in the evening between supper and bed.
2. The specific posture, or postures, you adopt are less important than physical stability. You want to do whatever allows your body to be at once relaxed and alert and thus whatever facilitates your being able to think about something other than aches and itches. The Yogis teach that one or another of the “cross-legged” sitting positions—such as padmasana and siddhasana—are the most conducive to unbroken concentration, whereas the Hesychasts make a practice of sitting on a short stool or bench, with their knees drawn up toward their chests. Simply standing is of course an option as well. Whatever the posture, you will need to make periodic adjustments—especially at first, as your body grows accustomed to the practice.
3. Calling it the Pax Profunda is a reasonably good way, and of course a thoroughly traditional way, of describing the operative or alchemical goal of any spiritual discipline—at least insofar as there is a “goal”. I add this perhaps surprising qualification because there’s a very important sense in which Prayer of the Heart has no “goal” other than itself. In Hesychast teaching, for example—and this is true mutatis mutandis for other forms of mantrayana—the Name of Jesus is sacramental in character, embodying (like the Eucharist) Christ Himself. In other words the Name is the Named, and for this reason the Prayer is an end in itself, or the End Itself. To invoke JESUS is already to be in His presence, hence in the Kingdom, whether you “feel” it to be so or not.
4. Discursive prayers, including words of praise and thanksgiving, are fundamentally cataphatic in character, whereas contemplative prayer is apophatic; the first corresponds to what we can say and think about God and is addressed to Him insofar as He is an interlocutor and agent, whereas the second corresponds to what we can’t say or think about God and is directed toward Him insofar as He simply is What is, at once higher than the Heavens and deeper than our own heart. These are by no means conflicting approaches, however. The Divine Essence and the Divine Person are not two Gods but one and the same Reality as seen from two different, but complementary, perspectives. In the words of Hermes Trismegistus, “It is not that the One became two but that these two are One.”