Some random thoughts:
I recommend that you discuss this with a professor who knows you well and show him drafts of your statement.
There is no reason why you need to submit the same research statement to every school. You can focus on different research topics, depending on the strengths of each department.
Your list of interests above is way too long and broad. I doubt it will be taken seriously. Focus on only one or two and discuss them in enough depth to show that you really know more than just the terminology.
I agree with everybody else that the statement should be no longer than two pages, no matter what.
Nobody expects an undergraduate to have much breadth or depth in their knowledge of mathematics. What you want to demonstrate is your desire and commitment to building greater depth in your knowledge of mathematics. Although you don't want to appear too narrow (and this does not seem to be a problem for you anyway), demonstrating breadth or an interest in breadth is far less important than showing the desire for depth.