Break up the search.dart code into manageable pieces

Right now it is a giant file with controller mixins, widgets, autocomplete and regular search logic, etc. This should be split up.

Start by reading search.dart and map its controller mixins, widgets, autocomplete logic, and regular search logic. Separate those responsibilities into manageable pieces while preserving the existing search behavior; done means the large file is broken up without changing functionality.

Scritto dal modello di indicizzazione a partire dal testo della issue.