Do mathematics researchers regularly solve problems like the ones from Project Euler?
This may be a better question for Math.SE.  That said...

These sorts of problems fall in the category of contest math.  Sometimes they are related to active areas of mathematical research (most often number theory, combinatorics, and geometry), but more often, current research deals with problems that are considerably more complex and can't be solved (or even stated) in a page or two.  

Note also that, at least in the US, there are very few people who get paid to do mathematics research full-time and exclusively.  Most professional mathematics researchers are professors at colleges and universities, and their duties also include teaching and administration (to varying degrees).

There is a recognized subculture of mathematicians interested in contest math.  They may get involved in creating problems, organizing contests, and coaching students.  At some universities, such activities may be considered a significant part of their research or "scholarship" duties, but they would normally be teaching regular math classes as well.

Edit: Actually, there is a category of careers I had forgotten: intelligence. The NSA and its counterparts (GCHQ, etc) employ many thousands of mathematicians. Of course, it's hard for an outsider to know what goes on there, but it could be that their activities have more of a problem-solving flavor. At least they are not bound by the requirement that their work be publishable! The NSA has a well-known (and highly competitive) summer internship program for undergrads, the Director's Summer Program, so that could be one way of testing those waters relatively early on.