I want to know a little about everything, not everything about a very specific topic. Is this a good/bad mindset for a career in research?
Instead of thinking about research as 'thinking up my own problems' - is there something that you read that your immediate reaction was 'that seems weird, I wonder why that it is?' and then not been able to find the answer? While it is true that research is about finding problems that other people haven't investigated yet, it is more true that research is about finding answers to those problems.

Reading widely is a good thing as it exposes you to different approaches and different potential sources of the question that sparks your interest. However, you are probably getting to the point that you should be focussing on a specific problem (since you have passed exams you must be some way through). I don't know what your program allows, but you might consider an interdisciplinary area where the balance between breadth and depth is different.