I have installed the Search API, Facets API, along with Search Pages. When the user goes to the front page, I want to display some default node results. How can I do this with just the node results for a particular view mode, without the user setting filter criteria?
I have very good experience in search api and the easiest way to display the search results is use module search_api_views (included into search_api).
Just create view for your search api index and select there required view mode and filtering criterias. Put this view on your front page and it will work.
You could do that by building a view based on search api index created for nodes.
enable the Search views module (search_api_views) a search api sub module build a view && add display page set the Default front page the url of the view page
Now you could have a front page from search api based on the index that you build before
the same thing will be done with module by adding pages
detailed tuts on search api will help you found here