Does the Search API module integrate with the standard search? I have Profile2 installed and want to be able to search on Profile2 fields that I have created. Does the Search API module integrate into the default Drupal search, which would therefore when configured, find Profile2 field values and the associated user?

If not, is there another approach I can use to include the Profile2 fields into Drupal standard search?
Search API and core search do not integrate. Just use Search API and scrap core search (Core Search does not work on Profiles as entities See http://drupal.org/node/1369544). The Search API can search on an entity known to Drupal. Profile2 creates profiles as entities. Therefore Search API can search Profiles. When adding a new index to Search API, just choose "Profile" or "Profile Types" under "Item Type".