Codex runs in the process list too: programs started by a Codex task are tracked like every other agent process, listed under running processes in Settings, and cleaned up when you delete the conversation or quit.
The phone's engine list follows the host: when you create a task, the engines you can pick (LEVEL11, Claude Code, Codex and your configured ACP agents) and whether each is ready come from the connected Mac/PC. An older host still shows the fixed three.