1. I am Bora, your student from last year.

Question

2. I am Bora, your student last year.

(Which one is correct?)

3. School begins at 8:30 am.
(What does school mean here? Does 'school' mean 'class'?)

Answer 1

Number 1 is better.

Yes, "school" means class. Most American elementary and secondary students stay at school all day so we speak of school starting at a specific time.