I need to know the differences in usages and situation for 学園, 学, 学校

So i am a bit confused on the instances of 学園(gakuen), 学校(gakkou), and 学(gaku) on their usage and how it might change the translation due to nuance usage such as name plates, grade, indication, print and speech.

I hope my understanding is correct but I may need some further clarification.

For one, I know that 学校 means school and 学園 means the institution of the school. But are each instances used only in specific cases. Like would one only be used to in speech and the other in text. What about signs that contain the name of the school?

Would 学校 only be used in instances to refer to only the grade level of the institution? But then what would be the purpose of just 学?

More examples of confusion and maybe each instance would mean something different or maybe would never be used this way at all.

中学
中学園
中学校

I don’t know if school name plates would even include the grade level in the first place (not counting colleges).

Another interesting point is for instance the name 私立石川学園 (Shiritsu Ishikawa Gakuendoesn’t contain grade level on a sign. Here is a rough translation is Private Ishikawa Institution. But it seems kinda redundant and I am not sure if I am using this properly since 学園 would make it redundant right? Or does 私立 determine that the ending is 学園 rather than 学校?

Then if we were to add 中 in there making it 私立石川中学園(Shiritsu Ishikawa Chugakuen) would make this wrong and by adding the grade level indicator i would need to use 学校 instead for a name plate? What about modes of speech or text used in a letter?

The nuances change right? Also wouldn’t change the translation as well to English? Because 中学 can mean both middle school and junior high. In English Middle school and Junior high school are relatively the same thing but I noticed that schools that refer them selves a junior high school rather than a middle school seem to be structured differently such as classes are similarly chossn like high school or maybe the level of classes are higher ,etc. So with Japanese I know that the different kanji for the different ways you can say school are actually important in context. But it’s a bit confusing. And I used chugaku as an example as it’s the best example of the level of education might something slight different depending on context and proper kanji.

So sorry for the long post but if someone who lives in Japan and can give me a detailed break down of the usages and translations of the different kanji for school.

I want to know the usages in real life, as manga and the authors don’t always use kanji or methods of talking in a realistic way that actual Japanese people use. I’ve been told that even kanji usages can be wrong too.

Thanks in advance and I figured this would be better as a separate post as the question is a bit too complicated to be put in the beginners thread as this whole thread revolves around context.