feat: add PostgreSQL database support

[chestercheng]

Summary

• Add the psycopg PostgreSQL driver and connection timeout handling.
• Add clean Docker-image integration tests for SQLite and PostgreSQL.
• Exercise webhook and authenticated dashboard APIs without direct database access.

Validation

• uv run pytest
• SQLite Docker API flow
• PostgreSQL Docker API flow
• uv run ruff check .
• uv run ruff format --check .

Depends on #11.

[iefiru]

LGTM

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 75.41%. Comparing base (065e3da) to head (fccd24f).

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #12      +/-   ##
==========================================
+ Coverage   75.31%   75.41%   +0.10%     
==========================================
  Files          16       16              
  Lines         717      720       +3     
==========================================
+ Hits          540      543       +3     
  Misses        177      177              

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