Why is it 我努力学习。instead of 我学习学得努力。? I thought it was like 他踢球踢得好。and not 他好踢球踢。(?)
Why is there 个子 in 他现在个子很高。? (I am also in need of examples that follows the exact same pattern like that). Should it rather be 他的个子现在很高。?
1) The meaning of "我努力学习" and "我学习学得努力" are not the same. One means "I strive studying" (I study hard) and the other means that you are learning to study harder (the positions of 学习 and 努力 are swapped between your first and second sentence).
2) "个子" signifies that you are talking about the person's stature ("he has now a very tall stature") You don't need the possessive "的", because the stature is not really "his" stature. He is just tall
Is 他现在身体不健康 another similar sensible example as followed in question no.2?
“个子高” is a set phrase used to express that a person (as opposed to a building or any other object) is tall, analogous to "背が高い" (literally "one's back is tall") in Japanese. So your sentence "他现在身体不健康" is correct in having no 的, but that has nothing to do with 个子, rather just customary Chinese sentence structure.
"我努力学习" and "我学习学得努力" are the same, meaning I learn hard.
"I am learning to study harder" would be 我正在学习如何学得更努力。Notice 如何 must not be left out.
As for 个子很高, other examples are 他地位很高，他血压很高（^=^）
It's the same reason you say "I try hard to study" and not "I study trying hardly". "try hard" is a verb, so you're using two verbs in sequence. The second example is putting the word in the place where an adverb would go, which is why it's 他踢球踢得好, because 好 is an adverb meaning "well" in this case.
There's a lot of instances in which 的 gets dropped off in more natural speech, this just happens to be one of them.