Backwashing will clear it out but if there is a problem with your filter, it will help find it and potentially help you get the pool clear faster.

The main problem with rain is that people don't go out and check their pools and thus chlorine levels get low. Just be sure you keep chlorine levels up. In my pool, a lot of rain will lower my pH.

You are right that bleach is sodium hypochlorite, usually 8.25% and yours may be 10 or 12.5%. (If it is at the advertised concentration although often that is not the case.)