In order to add more items to the search bar all you have to do is go to listings/attributes, then create a new attribute. In the box title where is says Search, check the Filterable box that says Display in the filter form. Then any attribute you added will be displayed in the search bar.
If you have the Bookings extension and features like multiple bookings per time period and time slots are disabled, you can add Dates to the search bar in HivePress > Settings > Listings > Default fields.
Also, if multiple places per booking and multiple bookings per time period are enabled, you can add the Guests field instead of the Dates field.
It’s possible, please try to enable multiple places per booking in HivePress > Settings > Bookings > Guests, * but disable the feature multiple bookings per time period, then search by date will be available.
As for special fields, you can create them using custom attributes and mark these attributes as Searchable.
Yes, please consider using categories for this purpose (another option is using a custom Select attribute and marking it as Searchable), especially if these listings can have category-specific fields (e.g. details specific for houses that shouldn’t appear for units).