Is there a way to have the unwrapping function not rearrange islands? I have a character I want to unwrap. I marked my seams and when I hit "unwrap" all islands are  auto-rearranged.

This is a great function but I need to overlap mirrored unwraps and the different scale and rotation of each island makes this difficult. Overlapping results not perfect as I would like. Is there a way to avoid this auto arrangement?
Yes - to some extent, you can pin UV's to control the placement of each island.
To control the scale and placement you only have to pin 2 uvs on the island, to control the overall shape better you may want to pin 4 or more.

See docs on live-unwrap and pinning, older wiki docs but still valid