I had the option to re-use the prod domain for non-prod a few years ago (the company's other two projects use the prod domain for all non-prod environments).

I didn't really think about cookies back then but it just felt like a generally bad idea because disastrously messing up a URL in some config or related service would be much easier.

Nah dev should probably be a separate tld so the cookies are completely isolated.

Stage, it depends - if you want stage to have production data with newer code, and are fine with the session / cookies being shared - host it on the same domain and switch whether users get stage or prod based on IP, who is logged in, and/or a cookie. That way your code doesn't have to do anything different for stage vs prod every time it looks at the request domain (or wants to set cookies).

If you want an isolated stage environment, why not just use a separate top level domain? Otherwise you are likely seeing yourself up for the two interfering with each other via cookies on the TLD.