Apache Solr Search Integtation and the Default Drupal Search Form
I'm assuming Drupal 7.


This is a limitation of the way core Search is implemented in D7. Try the Custom Search module; it can somehow change the way search paths are handled, although I have not used it.
One tab would be for core search, and the other for Apache Solr search. Go to /admin/config/search/settings and disable all search modules except Apache Solr, and also set it as default. Now the only available search path will be the one set up at /admin/config/search/apachesolr/search-pages.
I cannot replicate this on my Apache Solr &amp; Drupal 7 setup. In general, to change strings that are hard coded into modules, use the String Overrides module.