Full stack: Add the search functionality

Designs that should be available: https://www.figma.com/design/nlTRBsbrjta7EbmsNzAVdh/Notifications-mission--Dispatch-?node-id=2646-60840&t=fLyUyWzLVEHkPIKs-0

We should update the query params so that the search/filters is reflected in the URL so that the users can share filtered notification history views

Start with the notification history view and the linked Figma designs to understand the search and filter states. Trace how the view currently reads and applies filters, then define the URL query parameters needed to preserve them. Done means searching or filtering updates the URL and opening a shared filtered URL restores the same notification history view.

Written by the indexing model from the issue text.