I talked to my host about this problem prior to the update. My issue is that the form submissions aren't being emailed to me. However, according to my host, WordPress is using the host email, without credentials, which is denying messages from being sent. This wasn't a problem prior to my host cracking down on (potentially) unauthorized email from being sent through the server.
I've installed WP-Mail-SMTP, however it still seems to try to send from the wrong account. Fixing this issue would probably fix the other issues that are also listed above.