In Switzerland we use such name also for the "resident" permit of Swiss students. And now things are confusing: legal residence remain on old municipality (so canton must contribute to university), but the university city requires a permit. -- So maybe in such case it may just be the previous permit to stay on that municipality.

I do not think the question is about travels. expatriate SE may be more relevant (see links in top right icon). Do it matter the tittle? Considering that only 1 in 10 will be approved (there is a cap) your expertise, not the tittle, matter

What is the question? You get a refusal and a reason. Check what was missing or what could be misinterpreted. And be aware of scammers who try to sell "jobs".

You can say Turkey not counted as Asia, but the question is about genetics, so if the OP is a Turk, it may be on the the Central Asia region, or in any case more on Asian pert (Turkmenistan hints something, and also all languages of the region). Else.. it depends: Venetian origin? Greek? Ottoman? People/population is very different then geography.

OK, but to me the information makes things worse. In any case I wrote Canada, not UK, so it is much easier (for a Canadian resident, AFAIK) to travel to US, and for sure it is also nearer to parents). We cannot have all we want. I'm not Ronaldo with his skills (but I wish).

I do not understand why you mentioned "divorce" in the title. Why do you think it is relevant? To me it seems it can just add uncertainty (need to change live). Maybe she can do as Elon Musk: go to Canada (which should be easier of UK citizens), settle there for few years and later move to the US.

It all depends on I provided inaccurate information. We have no idea on how to assess that. Note: the visa refusal may hints they didn't interpret it as a genuine error but as an intentional fraud. (Note: they do interviews several times every days, so they are constantly trained, also note: Malta is in Schengen area, so information are shared with many other countries).

What did you did in the overstay time? Were you working? Attending an other school? Or just care your relatives?

Note: there is an international law about stateless people, which most countries agreed. A country should never remove a citizenship if such person will become stateless. Note: it may be a task of such person to notify this fact (countries do not know if you have other citizenship). Note: IIRC such law includes also rules about keeping residence as stateless (you cannot be move out of country (because other countries may not accept you). Note: Denmark signed it, as most of other European countries (but Poland and Estonia)

Note: this is a good question for the embassy. And you need to contact it in any case, to have the documents to be able to marry. (e.g. proof you are not already married, etc.).

Note: Applying to "green card" is a sign you want to stay, so the "overstay" is less a concern (you cannot overstay again with a green card). And it may depends if you were removed (with any charge). But if your daughter is not an US citizen, the I-130 backlog is years long, so at that time they may forgive you. But as Ozzy wrote: ask an immigration lawyer.

The last question: no. Where are plenty of PhD students from around the world (China has many of them), and it is not supposed that you travel every day or every week to your home country (and also Swiss students may travel on holidays on other continents), and that question is more suited for academia SE. University cannot help on visa (but on providing you good documentation). Just have plan B and plan C ready. Start collecting documents earlier, and prepare the application earlier (so you do not make the common last-minute errors which delay everything).