1. Fish Only With Live Rock
2. Both, the poop and uneaten food break down into oodles of organic compounds. Some feed beneficial bacteria like the nitrifying kind in your rock. Some in your tank, apparently, feeds some very, apparently, hungry algae.
the post on having two systems is likely right for you. I would suggest buying at least a 150 so the tangs can have some space, a 210 or 220 would likely please the tang police, lighting the FOWLR 220 would be exponentially cheaper than a larger reef.