You only capitalize 'student' in a sentence when it starts the sentence or it's part of a title: Students who join the Students' Award Committee will have their last class on Fridays free to...
At the beginning of a sentence or when using a name of a noun, and also when the word "I" is used.
There really is no specific reason. We just capitalize "I" even though similar words such as "you," "he," and "we," are left lowercase.
Yes. Always.
1. At the beginning of a sentence. 2. In the title of a book, film, article etc.3. When indicating a holy person or object.