Can I get an image containing both Ubuntu Desktop and Ubuntu Server?

Just an educated guess here since I'm not experienced with this sort of thing...but a possible solution could be to make your own ISO image which has the packages for both Ubuntu Desktop and Ubuntu Server included.

You could do this by using a program like RemasterSys to create an ISO from your hard drive after you installed all the packages you want your students to use, and then they can install Ubuntu using that. This way, only you need to download and install the packages once, and it also has the benefit that you can include or remove other packages as you see fit.

See How can I make a live CD/DVD from my harddisk installation? for more information on how to create such a disk image.

Simply creating a custom image based on Ubuntu Server with additional applications, including the desktop library components, will meet ALL your requirements. This is not both, but rather a "mash-up" of just what you need. The components would all be there after initial install.

I recently did this type of install to test out the HP MicroServer, but I loaded Ubuntu Server, and then added the desktop components as a command-line download.

Note: I would need to do some research to provide full details (if you require that), for creating the image.

Another alternative.

You could achieve the same result by creating a local cache server for Ubuntu updates, which would not require multiple downloads from the Internet (per student).

This would then allow unlimited alternative setups, all based on command-line recipes ..