If Visual Studio 2015 (and 2017) are installed from MSDN using a Visual Studio Professional ISO, can it later be upgraded to Enterprise using a new license key? ...or does it require a complete uninstall/reinstall of Visual Studio?
You need to install the enterprise version to obtain it, and VS cannot change the version through the type of your license key and the VS version depends on your installation.
For VS 2015, we can only keep one version on the computer. If you already installed the VS professional 2015, as far as I know, you can directly install run the VS enterprise 2015 installer to install it. After the installation is complete and successful, the professional is auto removed or replaced by the higher enterprise version. You can also try to use the forcibly uninstall command like: vs_professional.exe /uninstall /force to uninstall the professional version.
For VS 2017, please check the VS 2017 Release Notes and it describes Visual Studio Enterprise and Community (among other products) can now be installed side-by-side on the same computer. To make identifying these installations simpler, you can specify a short name or "nickname".