Is my tax return actually filed, even though it says I still need to mail it? I never filed them, but it says they are filed.

You must either efile your tax return or print and mail it yourself for it to be filed. If you chose to mail your return, it will always say Ready to Mail even after you mail it yourself because TurboTax doesn't know whether it has been mailed or not. Only you would know whether you printed and mailed it or not. 

Here is where to check if you efiled it: https://shop.turbotax.intuit.com/efile/efile_status_lookup.jsp

What *EXACTLY* are you being told has to be mailed? For example, when you e-file your tax return you will receive two physically separate e-mails. The first e-mail tells you that you have successfully SENT your return to the IRS. Then the 2nd e-mail will tell you if the IRS accepted or rejected your return. If you have the e-mail informing you the IRS *ACCEPTED* your return, then you are done.

It is also perfectly possible that with an e-filed return that was ACCEPTED, you may be required to physically mail separate documents to the IRS. (Such as a 1095-A for example) and the program informed you of this. If this is your case, then it's up to you to print whatever needs to be mailed and *you* mail it to the IRS.

If you elected the option to "print & mail" your return to the IRS, then it's up to you to print the return, sign it and mail it to the IRS. 

Under no circumstanceds and with no exceptions will TurboTax ever file a tax return for anyone by any means - electronically or otherwise.