Why does 学校 come before 年纪?
That's the order in Chinese and Japanese: school X学Y年. Not sure what the term is for academic year in university, however. Hanping suggests 年级.
The name for university is 大学, and my teacher said that you would say 大(number) for what academic year you are, like 大一，大二，大三，etc.
It's gotta be right
It looks to me like you have the wrong ”第” there. It should be "弟”.
Why is there not an 一个?
because indefinite article doesn't apply to chinese (but it's common in english)