Separate drupal view filters from the results
I haven't used the Drupal Search API but if you are creating a View with exposed filters, you can have them put in a Block instead of being part of the View.  Then you can put that Block anywhere you want.  To do this, you go into the Advanced Settings of your View:



and in Exposed form in block: select Yes instead of the default No.  Then you can go into the Block UI and place the newly created block wherever you want.

One interesting effect:  If you don't restrict this block to the page of the View in the Block UI, it will show up on all pages, and if your users then select any of the filters in it and hit apply, that will take them to the View page and results.