Unidentified Inductor type - opposing turn wound toroid?
All the turns go thru the core in the same direction, so electrically it is just an inductor, with slightly more interwinding capacitance then an RF guy would prefer (This is a low frequency application so the extra C probably does not much matter). 

It is probably done to accommodate sufficient turns while leaving a gap so the core can be epoxied to that mounting plate. 

The thin traces are probably provided so that at PCB manufacture an electrical test will catch some more etching problems (These will be more sensitive to over cooking it then the big copper areas), probably something added after they had a problem with a batch..... 

Regards, Dan.