Java

Have the students create a simple Minecraft mod.

If you're part of a large academic group they likely have a Microsoft contact that can hook you up with educational licenses, much like they do for office and other services.

anything that builds upon integration with external data sources and APIs, so JDBC drivers for example

I'd suggest studying the code of the Spring Pet Clinic webapp. It will teach you how to engineer a simple three tier webapp.

https://github.com/spring-projects/spring-petclinic

However this is heavily geared towards the Spring framework and not core Java. But this project showcases best practices for writing code and structuring a project.

Yes, loads; but without knowing what kinda stuff you want to learn to do, I wouldn't recommend anything in particular

Edit: might as well give some ideas though... Some libraries such as http frameworks will have some examples projects that use it, sometimes even in the core repo under ./examples or something. There are lots of small personal projects from random GitHub users that you can stumble across. Many useful and well written libraries are small enough to be approachable. But which of those to recommend depends on what you want to do with your code.