the proper way to offer condolences to a Jew

[Director Michael]

Shalom to respectful Rabbis and shalom Dr. Schulman,
As a Noahide, when a Jewish friend talks about a deceased relative and I want to show my respect, are there some standard words in Judaism, or are those words exactly like the ones other people use? (like G-d bless his soul or things like that)

During the first week that a Jewish person is in mourning over the loss of a spouse or close relative, G-d forbid, it is proper to include this phrase in your words of condolences, preferably as the concluding statement:
"May G-d comfort you along with the mourners of Tzion and Jerusalem."

After the first week of mourning, you can use any thoughtful and tactful words of condolences.

I have heard about "Baruch Dayan Emet", but then I heard it is said only when a religious person has passed away, or I understood it that way.

This means "Blessed is the True Judge," refering to G-d. It is said immediately by a person who hears of a tragic event, G-d forbid, and in particular when the person hears of the death of someone which touches them deeply - especially a close family member, G-d forbid.

After death what happens? Is the soul in another dimension and then it comes back to the corpse?

Please see our forum section titled "End of Physical Life" -
https://asknoah.org/forum/forumdisplay.php?fid=6