Direction of Study Suggestion I'm not sure if this belongs in stats or here, or why stats would be considered different to math. 
During my self-study of biochemistry and medicine, I notice that a vast range of published studies are flawed and irreproducible, leading to incorrect hypotheses about many things. In particular it seems the statistics in these studies, notably the P value and confidence intervals, are incorrectly determined and derived.
I therefore made the focus of my study temporarily on statistics, but I feel like I'm taking things on faith, such as Chi-squared tests etc, without actually understanding the mathematics behind the formula. I understand the law of large numbers, but not why a sample size needs n=30 or greater for this law to hold in some tests, and why not some other arbitrary number. I appreciate the integral of two points in a normal distribution represents the area, which is a reflection of the probability of an event, but this sort of explanation for more advanced concepts seems to be lacking in the intro courses. I don't like accepting such concepts on faith as seems to be the case with all the online introductory stats courses I can find.
I realized then that mathematical statistics is what I'm after, however my mathematical education doesn't extend beyond single variable calculus. 
I'm prepared to devote some time (6 months or longer) to understanding the foundational math necessary to approach an introduction to statistics from a more rigorous perspective. 
Would anyone be able to guide me on a study plan necessary that would lead up to probability theory and mathematical statistics, with an end goal of being able to fully understand applied statistics in science and the reasoning behind why certain equations and values of n are used.
Ideally I'd like to learn both the mathematical theory, and the application together, perhaps in a package such as R, where I can see the applications of calculus and probability theory to stats.
On an unrelated note, why are statistics and mathematics treated as two separate categories? I see stats often applies formulae, but not understanding the application of applied formulae from a mathematical grounding is surely a dangerous thing?
 A: I am a data scientist with an operations research and applied mathematics/computer science background. I agree that many people apply statistics in a "cookbook" fashion, without understanding the why...and it does lead to mistakes; however, science has managed to make great progress despite the difficulty in applying some of these statistical tools (and I'd argue that it is not success in spite of statistics.)
Now, as for a "study plan", if you really want to get a deep understanding of statistics, you need mathematical statistics, as you pointed out. And you will not get very far without two absolutely critical areas of math: Multivariate Calculus and Linear Algebra. You simply have to be comfortable dealing with multidimensional integrals, derivatives, and basic matrix operations and eigenvalue/eigenvector decompositions. No getting around this.
After that, I'd recommend you get yourself a good probability book to get basic probability under your belt (I'm thinking Sheldon Ross's A First Course in Probability or Jay Devore's "Probability and Statistics for Engineering and the Sciences").
Finally, dive into a nice intermediate math stat's text such as In All Likelihood by Yudi Pawitan (a personal favorite...and he uses R in his examples and on his web page!). If that looks like too much, Hogg's Introduction to Mathematical Statistics is also good.
That should set you up to be a much more sophisticated user and producer of statistics.
