Broad question! All I can offer is my own experiences regarding the above.
How closely related are math and statistics?
First of all, you need to make the distinction between applied math/stats and theoretical math/stats. I've found the applied/theoretical divide to be more substantial then the subject-level division per se.
For example, take a look at a graduate-level mathematical statistics textbook (Example). As you can see, it looks like it could be from a real analysis or measure theory course.
In contrast, applied math/statistics (my area) we are less concerned with the formal/theoretical properties as with their actual performance with real data. Hence, my concern is more with numerical/computational techniques (Monte Carlo, Bootstrapping, Numerical Integration/CAS, R/C++ etc), although theory is important too (albeit at a less abstract level, see example).
Can someone go from a math bachelor's degree (having taken some statistics courses) to a statistics graduate degree? Could someone who loves math also love statistics?
An emphatic yes to both! The only caveat is that most undergraduate math BS degrees are very theory oriented, so you may need to brush up on programming and computational math if you are going into an applied degree program. For theoretical statistics, you are all set!
What are the differences between how university statistics professors and math professors approach research problems?
Again, not much difference at the theoretical math vs theoretical stat level. Same thing for applied math vs applied stat.
How does the quantity and range of unsolved statistics problems compare to the ones in math?
Since "math" is such a vast subject, I think there are more unsolved problems in "math" vs just "statistics". However, both math and stats are coming up with new problems all the time, so I don't think you need worry about running out of problems.
As an aside, if employment is a concern, I'd look into the "big data" movement and the related techniques (data mining, machine learning, analytics). These areas are relatively new and there are lots of opportunities for contributing to their development (and potentially a ton of money to be made...think Google and the Cloud).