Shall I go and see my advisor's father who is on his death bed?

Unless you have some direct relationship with his father, this is not something we'd normally do in US, UK, or Australian culture. (I'm not sure if it's the same everywhere in the EU.)

A polite option might be to tell your advisor something like "I'm very sorry to hear about your father, please let me know if there is anything I can do."

If your advisor is from a different culture, that might change things.

The situation is the same in the EU as it is in the US. You should generally not go visit unless you are close to your supervisor's father personally.

If your supervisor has let you know about their father's situation, a reply similar to the one in Geoffrey Brent's answer is a good way to go. If you want to show polite interest, you can after some time casually ask about it, but quickly drop the topic if your supervisor replies vaguely or not at all.

I agree with other answers. Having said that, if the father does pass away, and if it is in the same town, you may want to go to the memorial service or funeral, as this is a way to show support for the living. Going with a group of students would also be fine.