That "hard green hair algae" sounds like turf algae, which is one of the best kinds to have in your refugium. It can absorb and grow on the tiniest amounts of nitrate and phosephate. But as I'm sure you know is also hard to harvest off the rocks. And if you are patient and want to save money, you could aquascape your new tank with dry base rock, and use your existing live rock to seed it. In 6 months it'll all be live rock. You just have to be very patient about stocking the tank till then. Phil
If you want to move the rock from your nano into the new refugium zone, that's fine. The sand should really be rinsed out to remove trapped detritus and get a fresh start. Save a cup of the live sand before rinsing to seed the clean sand once it is set up again. Algae on the rock will probably continue to live in the refugium. That's usually not an issue for most hobbyists. Any more rock you purchase should be fully cured, look clean and smell like the ocean (not black or rotten smelling).