Is it permissible to inquire about the circumstances of family members if that might lead to hearing bad news about them, and is that considered backbiting (ghibah), and is it obligatory for someone living abroad to refrain from asking about them?

Inquiring about family and friends is good, but if it includes a derogatory mention of the person being asked about, then it must be forbidden, because it is backbiting. And saying: "So-and-so was divorced, and so-and-so was evicted from the rental" is not backbiting, but mentioning the reasons might be backbiting if something is mentioned that the person dislikes being mentioned. So the rule is that whatever a person dislikes [being mentioned] is backbiting.