I don't think the Request Count goes up if there is an error during processing (though it may depend on what exactly the error is?). In my experience, if the email listener fails to process an email, the error count goes up and not the request count.
For 2) we don't store historical data about why an Email Listener's error count is going up. There should be an error in the logs when it occurs, but if not then enabling debug on com.pega.pegarules.integration.engine.internal.services.email.EmailListener may provide more information. If you're still not sure why the error count is increasing, I would recommend opening an SR for it to be looked into.
For scenario 1, does the email still process successfully?
For scenario 2, does the email fail to process? And does the listener continue to run, or did it stop?
My guess is that is the difference. In scenario 1, there was some non-fatal error, but we were able to continue processing and did. In scenario 2, some issue prevented the email from being processed and so the request count did not go up.
We are trying to identify which email is failed and why?
The debug output should tell you the subject and other data from emails that are processed. If one is throwing an error, you should be able to determine which.
If you're trying to find out which emails are increasing which count (request or error or both), I don't believe there is anything within Pega that will give you that data. An enhancement request would need to be entered for this to be considered for inclusion in a future release.