Consider a rename of the site to Transnationals. We could thereby expand the scope a little bit:
Questions about the transnational nature of the lifestyle of a person who has lived in (or lives in) more than one country;
Questions concerning multiple citizenship;
Questions relating to unions (marriages, etc.) between citizens of different countries;
Questions concerning the life with no citizenship, unclear citizenship, or a change of citizenship;
These categories are a little more broad than the idea of an "expatriate", who is someone who has simply left their original country. But the theme would remain the same. How to navigate life without the often assumed 1:1 relationship between a person and a country?
I'm not sure about the renaming of the site, but it might be worth discussing what we could add to our "what's on-topic" examples to explicitly cover the points mentioned by you. Let's do a breakdown though:
Questions about the transnational nature of the lifestyle of a person who has lived in (or lives in) more than one country - This is probably part of "Psychological and physical effects of the immigration to a different country", although it could be modified to say country/countries.
Questions concerning multiple citizenship - This is definitely on-topic and we already have a handful questions about this
Questions relating to unions (marriages, etc.) between citizens of different countries - This is one of the examples which is not fully on-topic based on our examples (as questions about short term marriage visas are more aimed for Travel than Expats), but most questions in this regard are definitely on-topic already.
Questions concerning the life with no citizenship, unclear citizenship, or a change of citizenship - This is also something we already have a few questions, and is generally on-topic already, as it's part of "living in a country as a non-citizen", but could make it more explicitly on-topic with some additional examples in our help section.
I'd support this proposal. My (very limited) experience of expatriates.SE so far is that nearly every question is about visas and immigration, and out of the two non-visa-related questions that I tried to ask, on one I was told "go ask on law.SE" and on the other there was total silence until I answered it myself.
At the moment this does not seem to be a useful resource for anything but immigration questions. Despite being a new expat, I've pretty much given up looking here.