If I have turbotax desktop, can I work on the same return online (and go back and forth?) - e.g. if I don't have my own computer on me all the time.

No.   The software is only on your own hard drive.  You cannot access the desktop download from a different computer.  

no. with desktop the only version of the tax file is on your computer. there is no copy of the file in your online account.  with online you can access your file from any computer after logging in. you can not access an online file with any desktop app unless you upload the file to the desktop. Then you can not download the file to your online account..