Is there a way to switch the Search form (used in the block) to do a Facet Search instead? I am using Faceted Navigation for Search and I want to switch the standard Search block to do a faceted search directly rather than the current situation where I have to switch to faceted search after the standard block shows up.

Is there a module for that, or can it be fixed somewhere in the options?
As stated below I originally understood your question.  Here is how you change the default search page to the one with faceted navigation:
In admin/config/search/settings you can turn off default node search and make faceted your default as shown in the below screen shot.


The Search API module allows you to choose other search methods, including using something like SOLR server for search, which also does faceted queries.