I'm currently following the story line in the Enigmatic Grotto (with the Undead being there as well), but I've been wondering something:
Is it necessary for completing the exploration to battle everything you encounter, or can I just retreat from everything all the time until the plot progresses?

I'm not sure, but in Astryl's Vale, you can retreat and still continue with the storyline, so I would assume the same for the other two regions as well. However, it can get monotonous over time, and it seems to progress faster when fighting anything in your path. At least, it felt like it to me.