Processes in Microsoft 365 for setting up Office apps, redeeming product keys, and activating licenses.
Yes, you should be able to install publisher 2013 alongside any Office 2013 bundle.
No you cannot use your Office product key to activate Publisher. Publisher will have it's own product key.
Note: when you change to Office 2016, you will have to change all of your Office applications at the same time. Many people have reported that the Office 2016/365 "Click to run" installer forces you to uninstall all applications from earlier versions of Office <sigh> (although one person said he was later able to re-install those older applications <yea!>