the easy answer about pumps, is if the pump runs and water flows, the pump is good; if not, then valves/leaks/clogs have to be found and fixed
Filter, there is no easy answer except to run it and see if the water starts to clear- it may take awhile, so no clearing doesn't mean filter bad immediately.
If you're familiar with electricity, replacing the timer is easy.
Start with what Ben said- add bleach and circulate the water (run the pump)... and maintain pH between 7 and 8...
We'll work on the other issues as you get that first priority sorted out!