I'll try my best to answer this:
What happens when a package is installed using a repository in Ubuntu
14.04 and the distribution is upgraded to 16.04?
Does the package remain functional or can it break (assuming the
package maintainers do not yet support 16.04)
The package will remain functional unless the new version 16.04 upgraded some dependencies (like libraries) that now causes the package to break or have weird behavior.
What happens during the update/upgrade process if the package does not
have Ubuntu 16.04 repositories?
If you try to update that package but the new version does not exist on any of your repositories, then the package will not be updated.
What if the package repository exists for Ubuntu 16.04 can it be
updated/upgraded as normal or are there additional steps involved such
as manual updating the sources list?
Usually it can be normally updated, unless the new version brings some new specific steps you have to do (like new dependencies that are not on the official repositories and such).