Update Windows 1.31.3.6868 to most recent version - says no updates available

Yes. The plex data folder is the same for both “bit” versions.

You can, but then your new server will have a new “identity”. Which means it may not appear in your Plex clients by default (but should be available under the sidebar’s MORE… mode), and the media shares you may have created for your friends & family will still point to the old server.
You do that by NOT importing the registry data of the old server.
You will very likely still have to perform Plex install fix for Windows Server (Claim issue)