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Can I use this plugin to authenticate a third-party app that doesn't rely on local folders?
No. That's not the purpose of this plugin. There are many extensions and libraries to share authentication between different platforms.
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Must the third-party app be part of the same website?
Yes, the plugin relies on same-site cookies.
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Can I install WordPress in a subfolder and still let it manage the main website?
Yes, WordPress can be installed in a subfolder and still handle the site as if it was in the root folder. See WordPress documentation for details.