When writing papers about new research made from my doctoral research results, after my defense, must my doctoral advisor be included as co-author?
The traditional role of authorship is to clarify who (significantly) contributed to the work being published. Doctoral advisors often contribute actively to the research being performed, and thus are included as co-authors on that particular regard.
However, they should not be included as co-authors automatically. Only if they have actively contributed towards the paper's ideas, implementation and/or writing. The least that can be expected is that they contributed to the paper's ideas (during your supervision) and that they have read and edited the paper being published.
That said, it is usually expected that you invite contributors to co-write a paper, if they have contributed significant ideas. If your doctoral advisor did that for the research you made during your PhD studies, then you should invite them for those particular papers.
After obtaining the PhD degree you're expected to establish yourself as a scientist who is capable of doing research on their own. Having your PhD advisor as coauthor to papers you publish after graduating runs contrary to this goal. If you did independent research as a postdoc, but add your PhD advisor as coauthor (by default or for some specific reason), then that could potentially reflect badly on you. The situation is hence different from a PhD student, for which it is probably beneficial to add their supervisor as co-author. In case you are accustomed to grant your supervisor co-authorship, whether it was warranted or not, you should now become less inclined to do so than you were previously since it might not serve your interests. Still, you can and should of course include them if it is warranted.
The answer is no, it must not, but it it ethically right to add him/her if any of the material has been discussed with or if you incorporated any suggestions from him/her. The list of authors of the paper shall be the list of people who contributed to the paper, and no person who contributed should not be left outside.
I don't see it wrong to invite your formed advisor to participate in your research. It would only be wrong for your former advisor to lead the research instead of you.
If the former advisor contributes in some significant way, then yes, he or she should be included. If no direct contribution is made then no. However, if this is an extension of your doctoral research then it would be quite appropriate to have your former advisor review a draft of the manuscript. This could help to prevent any unintended slight that might be perceived by excluding the name from the paper. This is always wise if the advisor is a big player in the particular area of research. If you remain in academia you will soon be seeking funding of your own and it helps to remain in the good graces of research community.