Do you know that when you save a file in 2016, you can't open it in 2013?
2013 is 32-bits and you have some macro's, then you need to modify them. I hope, that yo don't have VB6 applications with SW references.
You can't use SW2016 with 2013 license server and workaround with borrowing license in SW 2013 and use it in 2016 is neither working. Therefore you must upgrade license server first.
Yes, the server needs to be upgraded. Also Windows files.
As Ivana Kolin mentioned, be warned that once you have some people working in 2016 version then everything they work on can't ever be opened by those that are still on 2013.
First you do need to update your license manager on the server. This means :
- making sure nobody is running Solidworks
- on the server itself navigate to the "swlicmgr" folder and run the setup.exe in it. This will update the license manager.
- do keep in mind that there's a good chance that the server itself will need a reboot. In other words do make sure that the server isn't used for anything critical or that has an influence on other departments.
- Once the license manager is update you have to reactivate the licenses, this is an option with the manager. You go to the tab "Server Administration" and click on "Modify" and then select the second option : "Reactivate license"
more detailed explanation you can find here http://blogs.solidworks.com/tech/2014/10/how-to-upgrade-your-solidnetwork-license-server.html