How can I move the location of the advanced search form?
One option is the Advanced search block module which makes the advanced search form available as a configurable block. 


  Provides the advanced search form (normally displayed above search results) as a block. The block includes default values for the advanced search fields making it easy to refine searches.


From a brief glance at this module's code, it looks like it hides the default form and re-creates the advanced search form in a block.