no on paint! You will need to replace the liner.
Big NO NO ! It will soften or destroy the vinyl
High concentrations of chlorine can and will destroy a vinyl liner.
Yes, replace the liner. Nothing can put the color back into plastic or vinyl or glass or steel.
There are a number of ways you could remove stains from worms on a vinyl liner. You could use cleaners.
When you say a "pool liner" I am assuming you mean a vinyl liner as normally called a liner for short. If so then no, the liner cannot and should not be painted.
No. Replace the liner.