This is a excellent question and is addressed in many easily accessible books in English. A few worth looking at are, Modesty: An Adornment for Life by Rabbi Pesach Eliyahu Falk, in particular pp.228-235, Gefen Poriah: The Laws of Niddah by Rabbi Avraham Blumenkrantz, Chapter 2, pg 20, item 21 and footnote 68 and Family Purity: A Guide to Marital Fulfillment by Rabbi Fisher Jacobs, pp. 34-35 and 49-50. Rabbi Jacobs is a Lubavitcher Posek in Israel whose area of expertise is this subject. Another good source to look at is this link to the internet magazine, Halachically Speaking on the Shema Yisroel network. It is sponsored by the Kaf-K.
In answer to your first question, the practice is halacha, not minhag. It relates to two different areas. The first is a woman's obligations in regard to modesty. That applies whether she is a niddah or not. Some Poskim hold that women are obligated to cover their hair even in the total privacy of her home unless she has a specific need, like to shower. This is the preferred behavior according to Mishnah Berurah, Biur Halacha 75:2.
The story from Yoma 47a about Kimchit is in this context. That the rewards for modesty are great, both in Heaven and on earth.
The second consideration relates to Ervah, nakedness. A married woman's hair, when uncovered, is considered nakedness Mi'd'Rabannan. That also applies to her boys over the age of Bar Mitzvah and to her husband and the prohibition of making blessings or saying any kind of holy words, I.e. Kriat Shema, zemirot, saying a D'var Torah, etc.
These things apply to all Jews, not just Lubavitchers.
The second part of your question sounds like it is asking the deeper idea behind these laws. That is addressed in Sefer Ma'avar Yabok, Siftei Tzedek, chapter 15 which quotes the Zohar, parshat Naso at the beginning of pg 125b. These laws parallel a wife's monogamy and devotion to her husband with her monogamy and devotion to HaShem. It relates the covering of her hair to the concepts of not serving other gods and not following after the practices of the nations.
As to your third question about how strict this is, it states in Ma'avar Yabok that a woman not being meticulous in this area causes poverty to her husband and death for her children and that this is the decided law (דין). Poverty and death can mean many different things. They don't have to be literal. But none of them are pleasant. That sounds like pretty strict concepts.
Your comment about a woman's hair attracting kelippot is more relevant to a single woman, in particular a woman who is no longer married. That is a much more involved discussion, but should be understandable if one considers the differences between a woman who has never been married to one who is a widow or divorcee.
Your fourth question is more difficult to answer. In this day, the failure of someone to practice something is usually the result of a lack of knowledge, not because of rejection of the Torah. And that also applies whether a person is a Lubavitcher or from any other stream of practice and tradition.