Add 很 or not in these cases
Case one:

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

Case two:

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)