PhD in the US with an MSc from another country: coursework exemption?

It depends on the department. I have been looking at various applied mathematics departments in the USA and all of them want PhD students to go through the coursework and obtain an MS, even if they already did a masters degree before.

I think in general, there is no chance to skip coursework. This is probably because different universities have different standards and syllabuses for the same course. And they want to ensure everyone is up to the same standard and have taken the same syllabus.

But do check with the department, there are always exceptions.

P.S.

However, it doesn't have to be a "waste" of two years. The level of the course might be higher and you are probably able to choose different courses to get broader experiences. If all else fails and you are able to ace all the courses with no problem, you can always spend the extra time looking for a thesis adviser or find research collaborators.

Even at schools that have course requirements, you may be able to satisfy some requirements by taking a proficiency exam instead of an entire course. Check with the department!

The answer greatly varies. I was in the same situation, and in my school, a Phd student with a previous MSC gets exemption from 3 courses out of the required 12.

In other schools (I think it was Stanford, but I might be wrong), the required amount of (real) classes for a Phd student, is 4.

As @Legendre said, the best way is to check with the school itself. The Graduate Students Office (GSO) should be able to give you the details or refer you to the right website (yet, many times the website is not 100% accurate..)