Paste names, get a random gift-exchange draw where nobody draws themselves.
Yes. The draw is reshuffled until it forms a valid derangement, meaning no person is assigned to gift themselves.
It runs entirely in your browser and nothing is sent anywhere, but the result shows all pairings on screen, so draw it privately or have one person run it and message people individually.
Any number from 2 upward. For very large groups it still resolves instantly.