Students would need to subscribe to work with UE4 from home, and they wouldn’t be able to use UDK to create content for UE4. The two versions are not compatible in that way. You can read more about the specifics of educational licensing here:
I’ll try to answer your second question in more detail since it’s more in my wheelhouse. There are similarities in concepts and certain features between UDK and UE4, but workflows have changed quite a bit in many instances.
Trying to follow a curriculum based on UDK material using UE4 would probably be fairly difficult and confusing for the students. We basically ditched the entire library of documentation for UDK and created the UE4 documentation from scratch for these very reasons. In fact, we even had similar problems with people trying to use the Mastering Unreal books - which were written for UT3 - with UDK since the two versions of the editor were so different, and those were the same engine!!
On the bright side, we have been hard at work on that documentation as well as creating video tutorials, and there is a ton of example content available - the Content Example project is especially insightful. So there is a lot of content already available on day 1 to help both users and educators transition to UE4. And there is more documentation and more videos coming soon!
I hope that helps somewhat. If you have more question, please don’t hesitate to ask!