Certainly, in most countries. You and she are not biologically related.
Some religions and ethnic groups may assign a family relationship, even after the dissolution of your first marriage, and a few countries may incorporate such views into law, so you will need to assess the laws of the country where you live and the views of the culture in which you live.
In terms of genealogical relationships, a great niece is the child of one's niece or nephew, while a half great uncle is the half-brother of one's grandparent. Legally and socially, marriage between a great niece and a half great uncle would not be prohibited in most jurisdictions, as they are not closely related by blood. However, such a relationship may still be viewed as controversial or unconventional due to the significant age gap and familial ties.
yes
If the marriage was legal under the laws of the state in which it was conducted, it is legal in ALL states.
The niece could be the daughter of the sister-in-law's sibling and not related to you, so you could marry her. However, the niece could also be your mother or your aunt, in which case you could not marry her. Erm, but whover readwhat this guy just wrote, I expect you're wondering WHY WOULD YOU EVEN BOTHER? DON'T YOU HAVE ANYTHING BETTER TO DO??
No.
If you marry him maybe.
No
In most places, it is not legally permissible to marry your niece. Laws vary by location, so it is important to check the specific laws in your area.
The site stripped the parentheses from the question. It should read: "What is the technical term for the familial relationship you'd have with your half-niece's (daughter of half-brother) son?" Your half-niece is your half-sibling's daughter. So what is the term for your half-niece's daughter, in relation to you? My first best guess would be "half-grandniece", but that doesn't sound right to me, and I guess it with absolutely zero authority. Hopefully someone can give the correct answer.
If you marry your girlfriend, you niece (daughter of your brother or sister) will continue to be your niece and will also be considered by most to be your new wife's niece. Your new wife will be her aunt, as you are already her uncle.
He could not find another wife.
Yes. He had one biological child (Hailey), one niece which he adopted (Alaina), and also a half-brother which he is the legal guardian of (Nathan).