Depends on the university but often yes.

If Craig Wright has a PhD you should be able to find his dissertation, look up his advisor/thesis committee, see his publications in CS conferences, etc.

Perhaps this is it?

https://researchoutput.csu.edu.au/en/publications/the-quanti...

Wright, C. (2017). The quantification of information systems risk: A look at quantitative responses to information security issues Australia: Charles Sturt University

Doctor of Philosophy

Well if he submitted his thesis in 2017, he probably didn't have a PhD in 2015. Also there are several steps between submitting your thesis and getting your degree.

I suspect he may very well have considered himself 'basically done' with his PhD several years ago and just started using the title. He wouldn't be the first

Yep, that looks like it. And you can go from there to his advisor's home page and see he's listed as one of his PhD students:

http://csusap.csu.edu.au/~tzia/research.htm

Acknowledgments include supervisors/advisers.

A few weeks before bitcoin's release: https://archive.is/Ao7DQ

I doubt most schools would hire a teacher unless at minimum they are pursuing a master's/doctoral if they don't have one.

"Many 2-year community colleges often hire instructors with only master’s degrees, as long as they have some teaching experience. Four-year colleges and universities usually require a doctoral degree, unless you have decades of experience in your field, in addition to a master’s." http://www.teachingdegree.org/types-of-teachers/higher-educa...