As most of the probability theory results are extracted from measure theory which itself a pure Mathematics originated from Real Analysis.
No.
Probability can use a lot of material from a Real Analysis course. It doesn't have to.
Most MS programs in stats don't require Real Analysis. Some PhD programs don't require Real Analysis.
Something that would be useful is a course or two in graduate level Linear or Matrix Algebra.
Since nowadays there are plenty of statistical software, it is not essential to have a deep background knowledge of statistical concepts (but I say it is important). And I think its depend upon the research objectives, if it is a some common analysis problem which does not include any mathematical concept you will not need any expertise knowledge in pure mathematics but if the your research is theoretical one you will happy to see the joy of knowing pure mathematics. Also it is better to have some confirmed understanding of mathematics to follow research papers. So I think if you are a full time researcher in statistics you better have a knowledge of pure mathematics.
If you do not know anything about the central limit theorem and convergence of random variables, how do you assess whether you can use asymptotic tests for goodness of fit? And this is just an example.
My reply to the original question is: If you work in statistics, the more you know (on probability theory, measure theory, etc.) the better.
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