I'm interested in mathematical research, but want to know if there are good jobs besides being a professor. Are there any lucrative research-centric careers out there for someone with a PhD in pure mathematics?
I can speak from personal experience. I have a math Ph.D. on a topic with absolutely no real world applications, I work in industry, and my job title is actually "Senior Research Expert". So yes, there are research careers in industry. And while I won't get rich off it, I can't complain about the remuneration.
However, (almost) nobody in industry will pay you for thinking about stuff, going to conferences and writing papers. My research is extremely applied, and I am very much constrained by a) what customers will pay for and b) what is feasible given our code base and software architecture. I simply can't go off on a tangent and argue that a particular algorithm is very elegant, if it is not implementable in a reasonable amount of time or nobody will buy it.
Depending on your Ph.D. topic, be prepared to change fields. I used to do something like discrete optimization - now I do statistics and time series analysis.
While I personally don't code (which the developers are very happy about), I work closely with the developers that actually turn what I thought up into software. Interpersonal skills and the ability to fit into a team and a software development process (Scrum) are much more important here than in academia.
Research mathematicians are rare, and few people understand what I do and why I'm paid. It took a long time and some very fortuitous circumstances for me to demonstrate that I do add value to the company, and I was lucky to keep my research job when my company was acquired - my new employer really doesn't have research positions as such, so I had to justify why I should keep my niche.
Finally, there are about five other originally pure mathematicians in the team I work with, two of them with Ph.D.s. None of them do anything that could be called "research". They develop software or do analytics.
Summarizing: yes, there are math research careers in industry. They are few and far between. Be prepared for an uphill battle, to change fields, to do what needs to be done.
I can speak from personal experience from my job market experience this year. This year I'm graduating with a PhD in math. I mainly applied in academia, but I sent out a feeler for some industry to see if I found things I liked and in case I didn't get a job (maybe 2-3 applications). I am in algebraic geometry, and nowhere near direct applications to applied math or at least the definition that the USA has for applied math.
I got an on-site interview for a really great research job in the Defense industry. There was a lot of activity in my department with people interviewing with quant jobs on or near Wall Street. One student in my class took a startup job in the Bay Area. I got a job early on in the market in academe that I fell in love with, so I took myself out of the running and withdrew all my applications from industry and academia alike at the end of January.
A month later, Google headhunted me.
If you do good work and learn how to communicate quantitative ideas, then a PhD in math can really help you achieve industry goals. I think it is important to keep in mind that not everyone will get an academic job and a lot has to do with luck. I would not do a PhD in math if earning potential is your plan. A lot of these opportunities prefer a computer science background and will pay better for it.
That being said, to answer your second question, no one really speaks to you regarding salary until you get the offer, but in my interactions, the floor for starting salary for industry seems to be somewhere around 75-80k for these moderately prestigious industry jobs, and the ceiling seems to be around 150k. Results may vary depending on location and how valuable they believe you to be.
If you're interested in the job market for mathematics Ph.D. in the United States, a good start would be to take a look at the "Data on the Profession" section on the AMS website (see here). The annual survey features data on the salary of recent Ph.D. graduates depending on their area of expertise and the kind of work they're doing.
One potential shortcoming of the data mentioned above is that the statistics do not mention anything specific about what "kinds" of jobs the recent Ph.D. holders have in industry. To find out more about this, I suspect a good Idea would be to visit the websites of math departments, find their recent Ph.D. graduates and figure out where they work now. Of course, the majority of them will probably have research/teaching jobs in Universities, but I suspect a fair amount will be working in the Industry.
If you have a lot of time on your hands, you can always look up people on the list containing every (or probably almost every) Ph.D graduate in math in the US from 1999 to 2011 (see here) and hope that some of them maintain a LinkedIn profile where they report where they work.