It's interesting to note that ш, ж and ц are always hard consonants regardless of which vowel they're followed by. And one thing that и does, along with я, ю, е and ё, is mark the consonant that comes before them as soft (in much the same way ь does before a consonant or at the end of a word.)
As you can tell, we're in a bit of a softness paradox here. To solve this problem, the first rule supersedes the second: ши, жи, and ци are always pronounced as if they were written шы, жы, цы (even though writing the first two of these syllables is strictly forbidden by spelling rules.) It makes your observation quite pertinent, as an ы sound is exactly what an unstressed ей is supposed to sound like. :-)
It's the one oddity of the 7-letter spelling rule I just cannot make sense out of, but there you have it. More info on the topic here: http://www.russianforeveryone.com/Rufe/Lessons/Course1/Introduction/IntrUnit8/IntrUnit8.htm
Apparently "соул" https://ru.wikipedia.org/wiki/%D0%A1%D0%BE%D1%83%D0%BB
Useful hint: Go to the English Wikipedia page, scroll down and look at the list of languages on the left. If "Русский" is there click it to get the equivalent Russian page.
This subject matter should be in the spiritual skill and not in the body skill. The spiritual skill has words like religion, ghosts, and fate - beliefs and opinions like the concept of the soul. But the body skill is focused on words like fingers, legs, and headaches. It seems out of place here.
According to every major Russian dictionary the concept of soul/душа is in contrast with the body and specifically refers to religious conception when used in modern speech. The main concept this word carries is within the religious and spiritual domain, and thus should be taught in the section for religion and spirituality. Yes there are some idioms that use it as a metaphor for something you could consider secular, but even the metaphor relies upon the religious and spiritual understanding of the words meaning. But that's besides the point, most of the sentences in this course use the word literally.
Here is an entry from Ushakov's Dictionary (Словарь Ушакова), which is one of the major dictionaries in Russia, I've included a translation along with the original:
Soul: The religious and idealistic view - immaterial principle of life, sometimes contrasted with the body; incorporeal being that remains after death of the person.
Душа: В религиозных и идеалистических представлениях — нематериальное начало жизни, иногда противополагаемое телу; бесплотное существо, остающееся после смерти человека.
Take note of the "incorporeal being" part, incorporeal means not being composed of matter; having no material existence. That's the exact opposite of the concept of the material body.
Dictionary.com gives one English meaning as "the emotional part of human nature; the seat of the feelings or sentiments." Nothing religious about that, though it's not strictly 'body' either. That's the meaning most people will have in mind with a phrase like this.
To be honest, I don't get why this is something you care about so much.
This course teaches Russian, not English. Word concepts don't always perfectly overlap between languages, especially when dealing with abstract words. We are trying to learn Russian words and their meanings, so you should use a Russian dictionary. Don't assume that just because people have one conception of a word where you are from means that everyone else is the same - Russia is a much more religious country and puts an emphasis on the religious underpinning of certain words more than secular western countries.
To answer your question: I care because I'm passionate about education and learning, and these kinds of distinctions matter for the purpose of clarity, something we should all strive for when either educating others or learning.
We don't use the indefinite article "a" with "music" in English. You can talk about the music (for example, "do you like the music they're playing?") and you can talk about a piece of music ("Could you pick out a piece of music for radio show this morning?"). But if you want to talk about music generally, or as a subject, it takes no article at all.