If you put algaecide directly into swimming pool without mixing in a bucket of water first will it eat the paint away?

You don't have to mix the algaecide in a bucket of water, you can put it directly in the pool. I use Polyquat 60 that comes in quart bottles and add about 4-6 ounces per week. Just walk along the edge of the pool and pour it at the edge while walking. No algaecide will "eat the paint". It's not acid. In fact, all the algaecide does is strip the cell walls of the algae and make it more susceptible to the chlorine. It's the chlorine that kills the algae. However, one algaecide with a side effect of staining is Copper. It's a very effective algaecide, and it's safe - but it will stain concrete pools if it's used in a high enough concentration. And Copper can stain blonde hair as well. That's why I don't recommend using it. The preferred algaecide is Polyquat 60.