First up, please note that this Q&A is primarily opinion based. Therefore, I’ll be focusing on the symmetric part and nothing else you’ve mentioned and/or asked for in your question.
99% of cryptanalysis are not code breaker.
If you try a PhD in cryptanalysis and expect to break code, you are dreaming.
In movies such as Mercury Rising, Imitation Game or A Beautiful Mind, you see cryptanalyst (or child) breaking codes just by being exposed to the cipher text.
Academics consider this kind of attacks as weak. We now works something similar to a Lunchtime attacks. We consider the internals of the algorithm to be know by the attacker and the goal is now to find weakness in it. We are not looking for deciphering a single ciphertext.
Attacks that are used on these algorithm are far more complex (have a look at Differential cryptanalysis, Linear Cryptanalysis, Related keys...)
About the maths level, are required: Probabilities (a lot!), Finite Fields, Affine Varieties and other tools, all of them are about Bachelor-Graduate level.
What is the current state of the art?
In symmetric crypto, using the correct algorithm we are already post-quantum. Should you use strong enough algorithm, even the NSA won't be able to read what you are hiding.
In asymmetric crypto, the research is currently being done to get post-quantum assurance. We have things like New Hope.
EDIT:
As stated at the beginning of my answer, the question is off-topic by definition. The reason why it still exists and has been locked (by e-sushi♦) for “Historical significance”, is that we thereby gain a Q&A for all those newcommers to cryptography (and our site), who are confusing “fiction” (read: code breaking in movies) with “reality” (read: modern cryptanalysis in the real world). By keeping this Q&A around, we now have a place to point them to… hoping they learn something from it.
