Hi Mick,
There are no tools for backup in C++, you would be best looking at the mongodump and mongorestore applications.
As for gathering statistics you can normally run any command within C++ using the runCommand function, but without specifics of what you are looking for i cannot provide you with anything concrete.
As an example here is a short wrapper to get the dbstats of the test database:
BSONObj a = BSON("dbstats" << 1 << "scale" << 1 );
BSONObj ret;
c.runCommand( "test", a, ret);
Cheers,
David