Do all universities have a minimum required GPA for graduate school application?
I can't say "all" for sure, but I think you'll find that many if not most schools do have a minimum GPA requirement.

Whether you can compensate for a low GPA depends on how the school handles it.  In some cases it may be difficult, as admissions committees may literally scan apps for GPA and put all that do not meet the standard in a "reject" pile without even reading the rest of the application.  In other cases the minimum may be "soft" and they will read the app.  My advice would be to: A) get advice from a professor at your school you have a rapport with; and B) contact faculty at the schools you're applying to (or considering applying to).  I would not just bluntly ask the GPA question on its own, but email them in general saying you're interested in the program, and mention the GPA issue along with whatever else you ask them about the program.

You almost certainly cannot "throw away" your undergrad GPA in a PhD application.  Even if you have an MA, most schools will want to see all your college transcripts.

You should also take advantage of the time you still have during the year to bring your GPA up as much as possible.  Even consider doing this by taking easy classes that you might not otherwise take.  Personally, if I were in your situation, I would even consider delaying graduation by a year or part of a year, if it's financially feasible and your school will allow it, just to have more time to take classes and bring up your overall GPA.

You have already made a significant blunder in letting your GPA get that low, as a GPA below 3.0 will be a red flag to many graduate programs.  However, from what you say, you would be a strong candidate in other regards.  If you can get people to look at your app long enough to read your statement of purpose and see your grad coursework, then you probably have a decent shot.  To do that, you should leverage your strengths.  If you have taken many grad classes, done research, and published a paper, you presumably have good contacts in the physics department.  Lean on those people to write you glowing letters of rec.