How to transition from theoretical mathematics to theoretical computer science Background: I started my undergraduate career in computer science and transitioned to mathematics leaving computer science as a minor. I have just recently completed my Bachelors in Mathematics with a research project in Galois theory, algebraic geometry, and additive number theory. For the next academic year, I am pursuing a Masters in Mathematics which I will complete in a year through a 5-year (BS/MS) program. 

Although I did not complete more than a minor in computer science, I have been interested in having it play a vital role in my career goals in academia. 

My goal after finishing my masters is to transition into a Ph.D. program in mathematics, but I have not yet decided what I want to focus my research on.

I would like to learn about what Theoretical Computer Science is for academia and what background someone needs to become a theoretical computer scientist. 

Questions:
Do most theoretical computer science professors have their Ph.D. in mathematics?

How much formal background does one need from computer science itself? Can this be achieved outside of a school setting?

What fields of mathematics play an important role in theoretical cs?
The answer to your question in "no": Most of people who call themselves Theoretical Computer Scientists have PhD in (T)CS. 

And yes, you need to have a CS background because the word says it itself: Theoretical Computer Science which is the study of what a computer can(not) do and with what resources.

Math is a vital part and on different fields of TCS different fields areas of math play important role (number theory, linear algebra, combinatorics. probability theory are by far the most used ones). Generally speaking, the maths used are rarely "obscure" or too abstract/pure.

As on the question of if this can be achieved outside formal education the answer is "depends" on the motivation, circumstances etc. It is really hard to answer that.