An internship application letter should say what you want to learn and what you can already do. Employers reading intern applications are choosing on curiosity and coachability.
Nearly. 'Application letter' is the more formal term, common in South Asia and for postal or email applications. 'Cover letter' is the standard term in the US and UK. The structure is the same.
One page, three or four paragraphs. Anything longer is competing with your CV instead of introducing it.
'Dear Hiring Manager' is fine. It is better than a wrong name — and worth two minutes on LinkedIn to check first.