Old Salt Anchorman

Does this ability protect my minions from movement effects like forceful burrowing (e.g. Bury), submerge (e.g. Drown), and teleportation (e.g. Chaos Twister)?

Yes, all of those effects are movement. With Chaos Twister, don't even attempt to blow the card if the target is an ally nearby Old Salt Anchorman.

Does this ability prevent my opponent from rearranging sites? (e.g. Earthquake, Atlas Wanderers)

No, your opponent can still rearrange sites, because that doesn't count as moving the units, just the sites.