***How big is the 12.1.5337 package?***
Approx 80 to 90 MB
***If an installation package is assigned to a client group in SEPM, will the clients start downloading it as soon as I move them into that gorup?***
It's depend what setting you have done.Configure the Upgrade Schedule if desired or leave it unchecked to start the roll-out on the client's next heartbeat
***How can I tell when the clients have finished downloading the software from SEPM?***
Confirm that the upgrade completed successfully
You can confirm that the upgrade completed successfully by verifying the version number of the client software. The version number is displayed in the client's Help > About panel.
The Symantec Endpoint Protection Manager also displays the updated client version number on the Clients page after a successful check-in. You select the group, then the Clients tab, and change the view to Client Status
***Once the download has finished, can I move them back to their original group, or do I need to wait for the installation & reboot as well?***
You can you can wait the installation and reboot the system.
Upgrade Symantec Endpoint Protection 12.1 clients using AutoUpgrade
https://support.symantec.com/en_US/article.TECH96789.html