Do you capitalize scholarly?

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...

The word "have" is capitalized if it's the first word in a sentence, e.g., "Have you ever been to London?" If, however, the word "have" appears in the middle of a sentence, there is no need 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.