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/main.py

@@ -0,0 +1,37 @@
+from __future__ import annotations
+
+from fastapi import FastAPI
+from fastapi.middleware.cors import CORSMiddleware
+
+from app.api.router import api_router
+from app.core.config import get_settings
+
+
+def create_app() -> FastAPI:
+    settings = get_settings()
+
+    app = FastAPI(
+        title=settings.app_name,
+        debug=settings.app_debug,
+        version="0.1.0",
+    )
+
+    if settings.cors_origins:
+        app.add_middleware(
+            CORSMiddleware,
+            allow_origins=settings.cors_origins,
+            allow_credentials=True,
+            allow_methods=["*"],
+            allow_headers=["*"],
+        )
+
+    app.include_router(api_router, prefix=settings.api_v1_prefix)
+
+    @app.get("/", tags=["root"])
+    def root() -> dict[str, str]:
+        return {"message": f"{settings.app_name} is running"}
+
+    return app
+
+
+app = create_app()