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That error means that your API credentials are incorrect one way or another. You need to double check the values themselves, and also make sure you aren't setting up sandbox fields in the plugins with live credentials or vice-versa.
PayPal has a test server called the sandbox, and most plugins like that will allow you to setup both Sandbox and Live credentials. If it's set to Sandbox and you entered Live credentials then that would result in this error. Same the other way around, of course.
I would recommend you reach out to their support for help configuring their plugin. Again, though, it's just a matter of getting the correct credentials from your site.
If you're trying to set it up with your live account, just click this link and log in with your live PayPal account. It will send you directly to the API credentials page where it should display a username, password, and signature. Those are the values you would paste into the live fields for the plugin.
Angelaye is right in his assessment. (as usual)
To get an error 10002 the integration must be either:
1-wrong input in API credentials
2-Either pointing at sandbox with Live credentials or vice-versa.
Have you got any luck from the last response?