feat: add FastAPI backend with PostgreSQL and start script fixes

- Add server/ directory with FastAPI backend
- Fix server/start.sh to properly handle venv on Windows/Git Bash
- Add alembic migrations and pyproject.toml
- Add server tests

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

server/src/app/db/session.py

@@ -0,0 +1,44 @@
+from __future__ import annotations
+
+from sqlalchemy import create_engine
+from sqlalchemy.engine import Engine
+from sqlalchemy.orm import Session, sessionmaker
+
+from app.core.config import get_settings
+
+_engine: Engine | None = None
+_session_factory: sessionmaker[Session] | None = None
+
+
+def configure_session_factory() -> None:
+    global _engine, _session_factory
+
+    settings = get_settings()
+
+    if _engine is not None:
+        _engine.dispose()
+
+    _engine = create_engine(
+        settings.resolved_database_url,
+        echo=settings.sqlalchemy_echo,
+        pool_pre_ping=True,
+    )
+    _session_factory = sessionmaker(bind=_engine, autoflush=False, autocommit=False)
+
+
+def get_engine() -> Engine:
+    global _engine
+
+    if _engine is None:
+        configure_session_factory()
+
+    return _engine
+
+
+def get_session_factory() -> sessionmaker[Session]:
+    global _session_factory
+
+    if _session_factory is None:
+        configure_session_factory()
+
+    return _session_factory