European student trying to get a clear picture of US PhD system I am born and educated in a northern European country.

After I graduate I want to continue doing research within the topics I am passionate about. Much of the research I am interested in happens in the States. So, I am trying to get a clear picture of the PhD programs in the US. 

I have tried to find answers on my question online, but the university webpages something leaves me confused.  I would be very grateful if any of you can answer my questions or provide me with useful references.

Questions:


How does the application process work?
Do I apply directly to the university and the later if accepted look for a professor? 
Do I have to take courses before I can start doing research despite having a master and highly relevant skills for the topic in question? And if so usually how long?
How does research funding and salary work?
To answer your questions in order:

For academic graduate programs (that is, not a professional program like an MBA or JD), either at master's or doctoral level, you apply to the department via a central administrative unit which is usually called the university's &quot;Graduate School&quot; or something similar. The department website will probably have details. The contact person within the department/program is called the &quot;director of graduate studies&quot; or &quot;graduate director.&quot; It's a good idea, but not necessary, to contact potential advisors first if you have a clear idea of what you want to work on. Most American students do not do serious research-level work prior to grad school and do not have specific research directions in mind when they enter a grad program.
In addition to your academic credentials, you will almost certainly need GRE scores, and also TOEFL scores if you are not a native English speaker. Check with the program's grad director.

See 1.

Usually, but you can often place out of coursework by taking exams. It is unlikely that you will be able to avoid taking exams in the US system, regardless of your credentials. Check with the grad director.

PhD programs, especially in the sciences, generally admit students with a package of free tuition and a living stipend, which usually requires serving as a lab assistant or instructional assistant. Sometimes there is research fellowship money available through the school or through the advisor's grant which does not have these obligations. There are also some fellowships funded by government agencies, but these are generally reserved for US citizens. Master's programs may or may not offer funding. Check with the grad director and/or the university's Graduate School.