Hey, great questions! (and welcome to the forums) Here are my thoughts

1. Is each other a compound word?
I probably wouldn't go so far as to call "each other" a compound word, though they're definitely two words that occur commonly with each other (collocation). "Compound" usually refers to words formed by combining two words (e.g. "doorknob", "bingewatch", "Westworld"), though there are also compound nouns which is when two or more nouns appear together in a noun phrase.
2. What is the difference between a root and a stem?
The
root is the central, fundamental base morpheme of a word. On the other hand, the
stem is a base that other morphemes can be added on to. Probably easiest to illustrate with an example:
Let's take the word "holdings". The root of the word is "hold", with has had the morpheme "-ing" added. Then the stem "holding" has had the morpheme "-s" added.
So the root is the fundamental base, whereas the root is any section which a morpheme has then been added onto.
3. Is a free morpheme, in essence, just a lexeme?
So many free morphemes will be lexemes but the definition of the terms are different:
Free morpheme: any morphemes that can stand by itself (e.g. "dog", "Melbourne", "plant")
Lexeme: one of a set of words with the same core meaning (e.g. "eat", "ate", "eats" and "eating" are all forms of the same lexeme)