Cutter

Cutter's grappling can target objects. Does that mean she can grapple any edge of the map, like the water on the main map?

Yes. Those spaces are obstacles

Can Cutter's Primary Green land next to an immune hero? The card says "Move... to a space adjacent to an enemy hero..." — is the hero being targeted, or only the space? Can Cutter move next to the hero but not discard? Rowenna cannot use her Silver to get next to someone who is immune — do these play the same way?

No. There is a bit of confusion with immunity which refers to "checking presense" exception. It specifically cares about stuff like "if there is an enemy hero in radius..."