How can I detect what cookies a web pages uses?

This isn’t precisely what you’re asking, but I like using NoScript. It denies all JavaScript by default and allows you to whitelist by domain. It’s interesting to see how some sites load JS from dozens of different domains (Imgur, Politico…). One can assume that each of those sites is trying to use cookies and/or site data for tracking.

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whether they use cookies placed on the computer by other websites.

It should not happen on Firefox because of First Party Isolation. Basically it puts all the cookies created by a domain inside a virtual jar and makes the jar inaccessible by other domains.