To answer your questions one by one:
1. One page, font size 12, Times New Roman or a similar classic font
2. No matter your background, you always need to highlight 1. why you want to work in consulting, 2. discuss why you want to work for the specific firm, and 3. demonstrate your skills. I would recommend you to use a story-driven approach in your cover letter, using themes and stories to convey your experience and skills. In general, top-tier consulting firms don't care if you are a business major or something completely different.
3. Grades should be mentioned once IF you belonged to top 5-10% of your cohort. School can be mentioned IF you can build a story about your experience and skills that would appeal to consulting recruiters (see previous point)
4. Structure below
- Brief statement of intentions and personal contacts via referral if you have any
- Why you are the best candidate
- Why consulting
- Why the firm
- Brief concluding remarks
Also, be aware that the cover letter is not equally important across firms. McKinsey does not care much about it for instance.