Did you try unchecking login in options?

I already had that (first one) unchecked. What I'm saying is, if I uncheck the second one (Remember logins and passwords), I'm worried that Firefox Sync will not stay logged in, as the entries I'm seeing in my saved passwords (see below) would be deleted.

Ok what exactly is the problem? lol

Firefox will sync when it needs to, it doesn't "Login" or "stay logged in" to websites as a background process or something, You stay logged into websites by the cookies they force on yah and until they expire, you remove them or you're timed out/cut off on their server on their end.

Firefox always stores the credentials for Sync in the Password Manager, so if you remove these specific items then you will have to authenticate and enter the Sync password each time Firefox needs to connect to Sync. You should set a master password if you are worried about the security and when others have access to your computer.

• https://support.mozilla.org/en-US/kb/use-master-password-protect-stored-logins

If all I'm storing in the browser is my Firefox Sync credentials, I don't currently feel much need for a master password.

I'll explain this a second time: All I want is for Firefox Sync to stay logged in, but Firefox not to prompt me to remember my password on every site I log into (because I use an add-on for password management).

Yes, you can uncheck the box for "Remember logins for sites" and assuming you use a Master Password, when the Sync process starts you will still be prompted for your master password and Sync will still be able to use your saved login.

If in doubt, give it a try!