“arrive”是intransitive （不及物动词），不可以直接接名词的。

Yes it is. You need a preposition to connect "arrived" and "school". Without a preposition, you do not make clear the connection between the two words. While "at" is the preposition that everyone would expect (and it would be assumed if you just said "arrived school"), a number of others would work, including "from, below the, on top of. Some connecting term is required.

'At' is very necessary because it shows that the school is where you are arriving. If you do not use it, your sentence will not make sense. The sentence would mean that you are performing the verb on the school, or arriving the school.

Yes, it is totally necessary.

Yes, "at" is necessary. If it is not included, it seems that "school" is an object (it receives an action) when really it only tells where you physically were.