Add regression tests for content publishing and LLM timeouts

Clarify share file publishing boundaries

Add 62 tests across all planned phases + fix integration flakiness ...

- Phase 3: 19 API route integration tests (health, agents, sessions, messages)
- Phase 3: 7 WebSocket integration tests (connect, send, replay, invalid, stop)
- Phase 4: 9 agent tests (_check_context_size, _iter_stream_guarded)
- Phase 4: 5 planning tests (_parse_plan_steps)
- Phase 5: 22 tool tests (filesystem 13, code_exec 5, terminal 4)
- Fix flaky integration tests by patching module-level singletons
  (_session_store, _registries, _workers) instead of getter functions,
  because FastAPI Depends() captures the original function at import time.
- Update docs/testing.md coverage table (150 total tests)

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>