Hi Dustin, I don't see any reason why you could not use rock wool, sometimes I use coco coir. You can try to grow anything you want. People grow lettuce because the nutrients are generally low at first, lettuce, kale and greens don't require a lot of nutrients. So when fish are small and system is new its nice to see something growing half decent, and to have live plants in the system. Ya why not transplant things when they get big. My AP system is awesome for rooting cuttings and starting tomatoes etc. Haven't tried papaya yet but now that you mention it, I have a tough time getting them to sprout in dirt sometimes. But I'm not always good about watering them either. Potatoes haven't tried its too hot here I think, best to find seed potatoes to start with. Maybe a barrel with dirt and a drip from your AP. There's lots of options to explore and the ap water gives them a nice boost of fertilizer. Here's a link to a informative site that pretty much has it all.
http://community.theaquaponicsource.com/ 
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