[Frontend] Add search result accessibility and empty-state contract

## Problem
Pet and clinic searches need consistent loading, no-results, error, and keyboard semantics across pages.
## Acceptance criteria
- Search status is announced without interrupting typing.
- Results have stable keyboard navigation and selection semantics.
- Empty and error states explain next actions and preserve the query.
- Tests cover debounce, cancellation, keyboard navigation, screen readers, and slow responses.

Start by locating the pet and clinic search page entry points and inspect their current loading, no-results, error, debounce, and keyboard behavior. Add coverage for cancellation, screen-reader announcements, keyboard navigation, and slow responses; done means both searches preserve the query, explain next actions, and share the stated accessibility semantics.

Written by the indexing model from the issue text.