test: strengthen coverage for sessions, logging, and update check (#686)

## ℹ️ Description
* Strengthen the session/logging/update-check tests to exercise real
resources and guards while bringing the update-check docs in line with
the supported interval units.
- Link to the related issue(s): Issue #N/A

## 📋 Changes Summary
- Reworked the `WebScrapingMixin` session tests so they capture each
`stop` handler before the browser reference is nulled, ensuring cleanup
logic is exercised without crashing.
- Added targeted publish and update-check tests that patch the async
helpers, guard logic, and logging handlers while confirming
`requests.get` is skipped when the state gate is closed.
- Updated `docs/update-check.md` to list only the actually supported
interval units (up to 30 days) and noted the new guard coverage in the
changelog.

### ⚙️ Type of Change
- [x] 🐞 Bug fix (non-breaking change which fixes an issue)
- [ ] ✨ New feature (adds new functionality without breaking existing
usage)
- [ ] 💥 Breaking change (changes that might break existing user setups,
scripts, or configurations)

## ✅ Checklist
- [x] I have reviewed my changes to ensure they meet the project's
standards.
- [x] I have tested my changes and ensured that all tests pass (`pdm run
test`).
- [x] I have formatted the code (`pdm run format`).
- [x] I have verified that linting passes (`pdm run lint`).
- [x] I have updated documentation where necessary.

By submitting this pull request, I confirm that you can use, modify,
copy, and redistribute this contribution, under the terms of your
choice.

<!-- This is an auto-generated comment: release notes by coderabbit.ai
-->

## Summary by CodeRabbit

* **Tests**
* Expanded test coverage for publish workflow orchestration and update
checking interval behavior.
* Added comprehensive browser session cleanup tests, including
idempotent operations and edge case handling.
* Consolidated logging configuration tests with improved handler
management validation.
  * Refined test fixtures and assertions for better test reliability.

<!-- end of auto-generated comment: release notes by coderabbit.ai -->

docs/update-check.md

@@ -22,16 +22,15 @@ The interval is specified as a number followed by a unit:
 - `m`: minutes
 - `h`: hours
 - `d`: days
-- `w`: weeks
 
 Examples:
 - `7d`: Check every 7 days
 - `12h`: Check every 12 hours
-- `1w`: Check every week
+- `30d`: Check every 30 days
 
 Validation rules:
 - Minimum interval: 1 day (`1d`)
-- Maximum interval: 4 weeks (`4w`)
+- Maximum interval: 30 days (`30d`, roughly 4 weeks)
 - Value must be positive
 - Only supported units are allowed
 
@@ -93,4 +92,4 @@ The feature logs various events:
 - State file operations (load, save, migration)
 - Error conditions (network, API, parsing, etc.)
 - Interval validation warnings
-- Timezone conversion information
+- Timezone conversion information