Notifications of new e-mails not received when in screen lock. My understanding is that I should get a sound notification and a banner notifying me of the new e-mail.

Hello Iirish,

Being able to get the notifications the way that's expected, and that you have set is very important. With Mail you can have emails there is another setting that can influence the notifications. If you're getting the notifications after you open the Mail app, there's a good possibility that you may have Fetch New Data set to manually or set to Automatically. If you go in to Settings > Mail > Accounts > Fetch New Data, you'll see a variety of times frames towards the bottom, including the option for Automatically and Manually. If you have set to Manually you'll only receive those notifications after opening the Mail app as that would be when the email would be Fetched, and brought to your device. If you have it set to Automatically you'll only receive notifications as emails are received when the iPhone is connected to a charger, and Wi-Fi.

You can also find some helpful information in the following article.

If your Apple devices aren't getting Apple push notifications

Have a great week!