There are three unique words that begin with こころ～:
快い （こころよい）、 試みる （こころみる）、 志 （こころざし）
What is the origin of these words in relation to "heart/spirit/mind", if any??? Or is this just something coincidental? Why do only two of them have こころ in the kanji itself, but not the third??
By their meanings, I can kind of see how they relate to "heart/spirit/mind", but why are there only three such words that became so unique (I'm discounting ones like 心がけ、心構え、etc.)???
All these words are obviously based on 心, although they use different kanji today. You have to keep in mind that kanji is not natively Japanese, but rather a but rather a bunch of Chinese characters representing Chinese words with different semantic scopes than their Japanese counterparts. In the early days of Japanese writing, mostly everything that wasn't poetry was written in a style called kanbun, where it was practically translated into Chinese, and could be translated back into Japanese when reading.
You can read more about it in this thread:
Nuances between the different kanji spellings of あける：明ける vs. 開ける vs. 空ける
What happened with 快い et. al., is not different than what happened with many other words, such has 湖. I'll use 湖 as an example, since it's not abstract at all, and thus it's much easier to explain. ;)
In Old Japanese (and still today), the word for lake was just a compound of the word [水]{みず}, water (and more specifically fresh water, since salt water is [潮]{しお}), and [海]{うみ}, sea. Chinese, on the other hand, had an entirely unrelated words for "water", "sea" and "lake", and when translating to Chinese (for kanbun), Japanese writers had to use the proper Chinese character 湖 - they couldn't make a compound like 水海, since it didn't mean anything in Chinese. Later, the character 湖 stuck as the accepted character for みずうみ, although it was clear to anyone that this word comes from 水 + 海. For the very same reason, salt (塩) and salt water (潮) ended up being written with two different kanji although they used to be the same word.
According to the Japanese dictionary of goo, 試みる is literally "心見る". It is about checking or verifying (見る) the essence (心) of something. My guess is then that "こころみる" is a Japanese word, whose meaning in Chinese was "試". So, as for many other words, it's just a Japanese word on its Chinese character (which, even if it has the same meaning, may not have the same roots, hence no "こころ" in it.)
For the two others, well, 「心良い」is a valid spelling for one of the meanings (dare I say homonyms) of 快い meaning "good nature", and "志" is something you decided from your heart or a conviction.
I guess that's the answer to your first question.
The more you encounter vocabulary, the more you will encounter such words that embed another. I can't think of another example for こころ, nor any other "subword", but I can assure you there are quite a few, if not many.
