An internship motivation letter should be concrete about what you want to learn and what you can already contribute. Employers screening interns look for a specific interest in their work, not general eagerness.
A cover letter targets a specific job and its requirements. A motivation letter targets a programme or opportunity and focuses on your reasons and goals rather than your qualifications alone.
One page — around 400 to 600 words. European admissions teams generally expect brevity, and exceeding a stated limit counts against you.
No. The CV lists what you did. The motivation letter explains why it matters and where it is leading.