Write about coursework/background or research first in letter of intent? (and misc. question)
Put aside the details for a moment and think about the larger picture. You want your application to support your suitability for the program and your likelihood of success. Everything should support that and you have limited space to express it. For a doctoral program research is primary, so research experience, especially if guided by well seasoned and respected professors is a key message. Naming them is easy and may help and can't hurt. 

As to already being grounded in the required coursework, that is not trivial, but less important. There is time after you begin to decide which courses you need and which you don't, but having a solid background is positive. Your CV or transcripts will already speak to these, of course. 

So, the message is, I'm a great candidate, my background coursework and research has well prepared me, my interests and work habits will propel me and others can attest to my dedication and good results. Fit everything else around that message.