"Messages I send to my email alias don't appear in my inbox. Are they missing?"
SHORT ANSWER: Check Sent Mail or All mail. Nothing is missing. Everything is fine! :)
CHECK SENT MAIL OR ALL MAIL: For example, John's primary email address is firstname.lastname@example.org. He also has the email alias email@example.com. If John includes firstname.lastname@example.org as a recipient of a message, the message does not appear in his inbox. However, if John sends a message to his primary account (email@example.com), then the message appears in his inbox.
To prevent clutter, Gmail doesn't route messages that you send to your own alias to your inbox. You can find the message in Sent Mail or All Mail. If you need to see messages sent to your alias in the Inbox, you can configure the alias as an alternate "Send mail as" address for your account.