Your English is pretty good. It is clear. You've written a long, complex sentence. It is grammatical. A native speaker would make different choices of words and word ordering.
"Tickling" is not a good word choice. "Tickling" literally means this:
"Tickling" feels good and bad at the same time. It means something that makes you laugh.
"Nagging" or "bothering" would be better.
With some other small corrections, you could say:
"Here is a question which is nagging at me: at school, why were we told to write "have got" instead of "have" in sentences when, only now, I learn that writing "have got" is colloquial?"
The answer to your question is: schools are like that!