A motivation letter is what most European universities and Erasmus programmes ask for instead of a statement of purpose. It is shorter and more personal than an SOP — typically one page — and it answers three questions directly: why this field, why this university, and why you.
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.