I teach overseas, and I would never qualify to teach in an American school -- I have a college degree, and some teaching experience (from when I was in college), but no teaching credential.
An adjunct typically has a PhD or a masters degree combined with real-world experience.
Public school only require a bachelor's degree, but technically, teaching public school in most states also requires a teaching credential (especially with the No Child Left Behind law's "highly qualified standard), which is not part of most content (that is, non-teaching) programs. However, many districts are purportedly willing to hire a masters/PhD candidate and then back-fill coursework necessary for a credential (e.g., curriculum design, classroom management) as well as guide toward content certifications. These additional courses and certification exams are paid by the teach out-of-pocket, of course.
So, as with many things, YMMV.
No more than you are. If they don't have an M.Ed. or B.Ed. they're unqualified. Assuming you have those you may also need to do a certain number of teaching hours post qualification to get a teaching licence/QTS.
The typical adjunct almost certainly doesn't want to teach public school. Even in the crappiest community college discipline issues are minimal compared to teaching in high school or middle school and you have way more autonomy than a public school teacher.
With a masters you could teach in an American university, I believe the minimum requirement is a masters degree.