I think the practice is to put characters in the "New" section when they haven't appeared before, even if they've been mentioned.
As for reporting mistakes, there are a couple things you can do.
If you're fairly confident about a change that needs to be made, just hit the Edit button and make the change. If you're right, you have the thanks of other editors, and if you're wrong, you're unlikely to do damage that can't be undone easily.
If you're unsure about a proposed change, you can add to the Talk page for that article. (Hover over the button with the three dots, scroll down to "Talk", and click. Make sure to sign your contribution with the "Signature and timestamp" button next to the Italics button.
Some people use the comments on the article to point out issues. It would be better to use the Talk page, because the comments will be visible to casual readers long after the issue is resolved one way or another.
Thanks for keeping an eye out! We can always use a fresh set of eyes.