Why is 很 compulsory in case one while it is redundant in case two?
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的 is optional, you can say 他字写得很好，歌唱得很好，琴也彈得很好
很 is also optional, you can simply say 他字写得好，歌唱得好，琴也彈得好
的, and 是 are all optional, you can simply say 他字写得好，因為他比別人練得更勤力。
or, you can say 他的字写這麼好，是因為他比別人練得更勤力。
You find 他的字写得很好，是因為比別人勤勞練字的關係。awkward, because the better adverb in this context is not 很(very) but 這麼 (this)
他字写得(這麼)好 ... = His penmanship is this good because... (這麼好 make the sentence sounds incomplete, therefore, point to the need of a cause)
他字写得(很)好是因為... = His penmanship is very good because... (很好 make the sentence sounds completed, therefore, doesn't point to the need of a cause as much)
很: (i) is often used for balancing (e.g., 很好), so doesn't necessarily carry its full weight (cf. English, when you say "I'm very well", in answer to "How are you?", where you don't necessarily mean VERY);
(ii) is an absolute (vs relative) value, so no contrast/comparison implied (这个很好 is an absolute statement, just commenting on "this"; vs 这个好 is a(n implied) relative statement, sort of "THIS ONE's good [implied: vs that one]"). This is why absolute 很 cannot be used in a 比 sentence.
Your examples: Case 1 他的字写得好。（✕）--> There's nothing wrong with this, if the context implies a contrast (他的字写得好, vs his singing / cooking, say, or vs someone else's 字)
他的字写得很好。（〇）--> This is just an absolute statement, so no implied contrast with his other skills or someone else's 字. (But can be used as an implied contrast -- need context; "rules" are never 100%, remember.)
他的字写得好，是因為比別人勤勞練字的關係。（〇）--> No absolute 很 because it's a comparison: His characters are better because...
他的字写得很好，是因為比別人勤勞練字的關係。（✕）--> This one is acceptable, actually, although the second half has a comparison, because it could be that his characters are (very) well written because he practises more. Reason: the first half is a statement on its own (not a comparison with the second); the second half is the reason for why his characters are well written; so it's not a direct comparison as such (cf. Can't do: 他的字写得比别人很好 where absolute 很 and the comparison are sitting in the same sentence).
Both sound natural to me.
sounds a bit odd, because the two clauses states different aspects. 他的字写得很好 states his writing is quite good, which sounds a complete statement and doesn't need a reason. But others might have different opinions.