Is it possible to move names and parts separately in board edit mode in Eagle?
Yes, it's called the Smash feature. Select Smash from the toolbar, click on the part, and the name and value labels can be moved and rotated independently from the part itself (with their own origin).

When moving the name or value labels after smashing, a helpful line appears that shows to which part the label belongs. When moving or rotating the part itself, the name and value labels will move and rotate the same relative amount so they don't become visually disassociated from the part.