Does it ever make sense to license source code as a learning resource under GPL?
Of course it does. Developing a learning resource takes time / effort / money, and someone doing it is perfectly justified in placing conditions on how it is used. Using GPL is effectively making the resource available for free ... with certain limitations that are designed to further the aims of building "the commons".
Anyway, is this a standard practice at all?
It is not standard practice to use AGPL. But it is common practice to use GPL and/or one of the Creative Commons licenses.
I realized though that if I copy and paste a single line of code, my project would become AGPL. Possibly by even more trivial actions? Am I just being over-paranoid?
I don't think you could be "dinged" for copying and pasting a single line of code. If it came to a court case, you could argue that copying a single line was either fair use or de minimis; i.e. too trivial to justify any damages. No sensible judge would allow a lawsuit based on one line of code to proceed.
However, it is clear that fair use or de minimis do not allow wholesale copying.
Also, what benefits are there to this licensing scheme?
AGPL is essentially GPL taken a bit further. The idea of GPL is that you should be prepared to "give back" to the open source community any software that you develop based on the GPL'ed code. The benefits (to the community) are manifestly obvious.
AGPL takes this a bit further by including material (e.g. your content) that is delivered to end users via a web server as being affected by the license. I don't know the circumstances, but it sounds like using AGPL might be a reasonable thing for them to do.
Either way, it is not your place to implicitly criticise them for how they choose to license the results of their efforts. If you don't like their conditions, don't use their IP.