screwball, my tank is too new to have algea. I couldn't think of the word at the time I typed here but it actually was calcium and limescale deposits. Water hear in Florida is hard.
I'm in the middle of cycling my own 10 gallon tank!
You could use the old filter but it won't filter that amount of water properly. But it should do till you get something better. Just as long as you didn't use any cleaners on it. I read hear that media from an established tank is actually good in helping your tank cycle faster. Just as long as you didn't throw away any of the media and also if you didn't let it dry out. That's what I read so far. We are almost on the same boat.
When I'm done cycling this tank I'm going to use the media and substrate to help me cycle my 25 gallon tank.
Glad your Betta has a bigger home.