I've had a single SAE for almost 3 years now and it is 4". I've never had black beard algae but I did have green and he ate it. I have read that they do school but my single seems to be doing just fine in a 55 gallon. He tends to "school" with the gold barbs.
If you haven't yet started with cycling, you can fill the new filter with seeded media from your 20gal and right away add your betta and shrimp at the same time. The seeded media gives you an instant cycle.
You can't leave the seeded media in an empty tank for more than about 6 hours or the BB...
When I was trying to get rid of my snails that came in on the plants, I would check just above the water line in the morning, squish them and drop them in the tank - my fish went crazy for them and ate every one I dropped.