Can I move this stake?
Yes. It is a movable obstruction. See Rule #15.2. I can remove this if it impacts my swing or stance. Note that tee markers are not movable obstructions on your tee shot, but they are movable obstructions on any other shot.
While we're doing pictures, is this ball embedded?
Nope. It is very muddy, though. I thought it was embedded when I hit it, but it bounced out of its pitch mark. I must play it as it lies. I hit a pretty good chip (for me) and then three-putted. :|