It seems a little cumbersome to say I am not at the house but at the school. I would think that "I am not at home but at school" sounds more natural, but maybe the Hungarian sentence would be written without "a" before háznál and iskolánál in that case.
There is a semantic difference between "at school" and "at the school". The first implies belonging - i.e. you are working/studying there - while the second simply that you are at that physical place. Similar distinction between "home" and "the house"
Oh, no no, at least not here. "At home" - itthon/otthon refers to being inside of your living place. But here you have a háznál - by the house. You're outside, maybe waiting for someone to pick you up.