Let me explain the relation between Ki and TKD.

At first, I think we should define what Ki is in the relation to Taekwondo, in order for credible discussion.

I suggest the concept of Ki that "Ki" is a flow of vital power. It can be perceived subjectively with ease, and it can be perceived objectively only in a rare case that Ki is strong enough.
(Let me confess that I don't know exactly what Ki is. Perhaps no one knows what it is exactly. In this discussion I want to take some part in your study.)

At the beginning, I think, we'd better exclude the mysterious cases like that a man strike his opponent from 10 m far away with his palm blow like a air gun, although it exists in real.
(I know such a case in Korea. It was reported in famous and credible news magazines.)

Then we think that every vitality exists only in living thing. The living has its spirit and body. Then, the Ki as a flow of vitality can be created with the cooperation and harmony of spirit and the body. If you can regard my short discussion as reasonable, you can understand your experience of Ki in this conception.

The relation of Ki and TKD is evident, then.
TKD requires you to train your mind and body together, not only in the same process, but also in the harmony of both. TKD's motion can be itself without your spirit needed in it. How to get the harmony of mind and body? It is to get the essence or the core of life; the breath. (See chapter 14 of PPT)

Now we can recognize the relation between breath and Ki. According to experience and my knowledge, every training and process to build and control Ki is the way of breathing. In TKD breathing is also essential. Particularly, controling mind and breathing are main factor of controling attack and defence. (See chapter 38) "Breath is a long joint of the change extended from the ceiling to the bottom of phenomena of life, and an axis that binds mind and body to oneness. To control this breath is to control the mind, and then, the motion..."(chapter 38)

Every sport or every motion needs your proper spirit for the action. However, the spirit itself is not more important than the motion. In TKD it is. Some beginners come to face difficulties in their training because they focus only on their motions, where they have to control and move their spirit. For example, they just kick or punch without any intention. Whenever I find them, I point out the fault. This reason generates the a rare case that the high level TKD man surpass and control the opponent only with his Ki so that the opponent cannot even try attacking him and give up fighting.

In Ki-Kung, great parts are controling the mind and breath. There have been an important Ki training, called "DanJeon breathing", a traditional process of Ki training of Korea. (I also trained it in my childhood). It builds up my health and even cures my sickness.

Yoga, the traditional Ki Kung of India also has the breathing method as its essence. It is important to control the breath at first in any kind of Ki Kung, then to control the mind. We have to control the breath and some Ki produced by breath with our intention. We can find that the Ki is another aspect of the mind. Only the thouth that I move Ki to this way moves the Ki to this way actually. Whenever I got some idle thought, Ki is scattered right away.

In TKD, everyone have to learn how to move in attacking and defending at first, then he or she have to learn how to control his or her mind. Especially in Kyorugi. The formal structures are same between Ki Kung and TKD. But there is difference. TKD focuses on its moition of attack and defence, but Ki Kung on the flow of Ki. The difference is so trivial that Ki is very negative motion seen outside but very active motion seen inside while the motion is very negative flow of Ki seen inside but very active flow seen outside.

I can add some more comments to this, but they would be somewhat repeating same concepts. So let me finish the answer to the first question. This answer is not the perfect one, as I've emphasized at the beginning, and I hope another get engaged in this discussion.