Hi Pierre,
Unfortunately PS3003 is a generic name used for various low cost power supplies, and bad caps keep their pics behind a 'registration wall', so posting your own pics will help establish the identity of your supply. That said:
1) difficulty to adjust the voltage maybe due to a faulty potentiometer - sometimes just excercising the pot may improve its behaviour. If that doesn't work then probably best to replace it.
2) Could be same as 1), but not clear what the exact symptoms are. Are you saying that turning down the current limit has not effect? Did you try with the output shorted?
3) Important to know what model you have, but typically, relay switching point is based on output voltage, not load, so no need to have anything attached to hear them click.
As far as lithium ion battery charging goes, I'd highly recommend that you make sure your that battery has its own protection circuit. You dont want a spontaneous fault or accidental user error with the power supply causing a fire.