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/api/v1/endpoints/health.py

@@ -0,0 +1,34 @@
+from __future__ import annotations
+
+from fastapi import APIRouter
+from sqlalchemy import text
+
+from app.core.config import get_settings
+from app.db.session import get_engine
+
+router = APIRouter(prefix="/health")
+
+
+@router.get("")
+def health_check() -> dict[str, object]:
+    settings = get_settings()
+    database_ok = False
+    database_error = None
+
+    if settings.setup_completed:
+        try:
+            with get_engine().connect() as connection:
+                connection.execute(text("SELECT 1"))
+            database_ok = True
+        except Exception as exc:  # pragma: no cover - runtime connectivity branch
+            database_error = str(exc)
+
+    return {
+        "status": "ok" if database_ok else "degraded",
+        "database": {
+            "configured": settings.setup_completed,
+            "ok": database_ok,
+            "error": database_error,
+        },
+        "redis": {"configured": bool(settings.redis_url), "enabled": bool(settings.redis_url)},
+    }