"I am not a middle school student."
As a general rule Japanese individuals do not use subjects in their sentences so long as that subject has already been established. For them it's the equivalent of point at something and calling it out, which is considered rude.
As a foreign speaker they will not think less of you for always using a subject, or referring to yourself. However, they will be more impressed if you know the appropriate time to use a subject and when to drop it.
The rule is:
1) If the subject is understood or previously established do not use a subject.
2) If the subject is not understood, or is being changed use the subject once and precede from there.
Yes and no.
There are many formal instances where using わたし would be considered rude. In a business encounter if you used わたしit would be taken as being prideful. When 'I' is the understood subject being spoken about you do not need to constantly be referring to yourself.
Say you were at an interview and describing your skills. A constant repetition of わたしwould be taken as boasting about your skill when ideally Japanese prefer to come across as humble and not putting too much of the spotlight on themselves.
Again, this is something they won't hold against you very much when you're a foreign speaker. They make a lot of allowances and in my experience are just thrilled you can converse with them.
わたし, and how to use subjects in conversation is also a tricky nuanced skill to develop and not one to worry about too much at the start of learning a language. It's one you'll get more of a hang for the more you interact with native speakers.
中学生【ちゅう・がく・せい】… ～生（せい）a suffix that denotes student
年【とし、ねん】Usually ねん when denoting year as a suffix. It's not used in this sentence.
You can say 私は中学生いません but that most probably will be understood as simplified version of「私は中学生がいません」"I have no middle schoolers".
です and いる work differently, your confusion probably comes from Duo teaching ではありません which is a construction that negates です most of the time. So you can see ではありません、ではない、じゃありません、じゃない as the negative counterparts of です which usually involve a noun that connects to the whole structure.
いません in contrast, is the masu polite form of いる、a verb, this is very similar to "being" in English, but not quite. It's also used in other ways, the same happens with ある・ありません
So in short:
「中学生がいません」"Middle schoolers don't exist (here)" or "I don't have middle schoolers".
「中学生です」"I'm a middle schooler".
「中学生『ではありません』」"I'm not a middle schooler".
「中学生『じゃない』」"I'm not a middle schooler".
「私は、中学2年生じゃない -です」"as for me, (I'm) not a middle schooler in second year". The last です in this sentence is added to make the sentence sound more polite, is not like the other です。I know, confusing isn't it?