SearchAPI & FacetAPI: search box above I need to have layout like this:
[search box]
[facet - taxonomy 1]
[facet - taxonomy 2]
[search results]

But there is no block for the search box itself and no block only for the results.  Or can I somehow manipulate it in the template? I'd rather not use JS to move the components around.
You can use a view to display the search results, and exposed filters for the search box.
You should set exposed filter of this view as block (in the view settings). Then you can set the form above the search results.