Answer:
It is not normal for email messages to show up in your "Sent Mail" or "Sent Items," unless the email messages were sent by you.
But, this is what may have happened to you. If a spammer/cyber-criminal sets an email message's "From" address the same as the "To" address (in this case your email address) Google/Gmail doesn't filter it as spam; instead, it sends it to "Sent Mail" automatically, even though you didn't send the message.
You can set up a filter to delete all messages that have the same "To" and "From" address, but that will mean you will not be able to send yourself an email message, because the filter will delete it. But, use the filter until Gmail fixes the problem.