refactor: align backend layout and supabase infra

Consolidate backend modules/tests under the backend package while syncing Supabase compose/env config and related plans.

backend/tests/e2e/test_mobile_health_e2e.py

@@ -0,0 +1,57 @@
+from __future__ import annotations
+
+import socket
+import threading
+import time
+
+from playwright.sync_api import sync_playwright
+import uvicorn
+
+from app import app
+
+
+def _find_free_port() -> int:
+    with socket.socket(socket.AF_INET, socket.SOCK_STREAM) as sock:
+        sock.bind(("127.0.0.1", 0))
+        return sock.getsockname()[1]
+
+
+def _wait_for_port(host: str, port: int, timeout: float = 5.0) -> None:
+    deadline = time.time() + timeout
+    while time.time() < deadline:
+        with socket.socket(socket.AF_INET, socket.SOCK_STREAM) as sock:
+            if sock.connect_ex((host, port)) == 0:
+                return
+        time.sleep(0.05)
+    raise RuntimeError("Server did not start in time")
+
+
+def _start_server(host: str, port: int):
+    config = uvicorn.Config(app, host=host, port=port, log_level="info")
+    server = uvicorn.Server(config)
+    thread = threading.Thread(target=server.run, daemon=True)
+    thread.start()
+    _wait_for_port(host, port)
+    return server, thread
+
+
+def test_mobile_health_e2e() -> None:
+    host = "127.0.0.1"
+    port = _find_free_port()
+    server, thread = _start_server(host, port)
+
+    try:
+        with sync_playwright() as playwright:
+            request_context = playwright.request.new_context(
+                base_url=f"http://{host}:{port}"
+            )
+            try:
+                response = request_context.get("/api/v1/health")
+                assert response.status == 200
+                body = response.json()
+                assert body["status"] == "ok"
+            finally:
+                request_context.dispose()
+    finally:
+        server.should_exit = True
+        thread.join(timeout=5)