Since the properties of (traditional) magnets are dependent on the crystal structure and atomic properities of iron, do these properties remain once the iron is molton?

Nay. At the Curie point it loses its ferromagnetism. Iron melts at 1811 K and its Curie point is at 1041 K.

 

 

From a post on this forum, in December 2008 by Gilded

You should study ferrofluid magnets.

 

http://en.wikipedia.org/wiki/Ferrofluid

Ferrofluids will still fail above their Curie point; they are colloidal fluids, not a pure liquid.