Is Codex ok with any other third party applications, or just those?

Yes. You can build third party applications on top of codex app server. All open source. https://developers.openai.com/codex/app-server/

  Codex app-server is the interface Codex uses to power rich clients (for example, the Codex VS Code extension). Use it when you want a deep integration inside your own product.
It mentions 'Inside your own product', but not sure if that means also your own commercial application.

I think it's permissible. Zed uses it to power their Codex integration. OpenAI has been quite vocal about it.

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By default, assume no. The lack of any official integration guide should be a clear sign. Even saying that you reverse-engineer Codex for apps to pretend to be Codex makes it clear that this is not an officially endorsed thing to do

Codex is Open Source though, so I wonder at what stage me adding features to Codex is different from me starting a new project and using the subscription.

But I believe OpenAI does let you use their subscription in third parties, so not an issue anyway.

Interested to know this too