[jfitzsimons]

There have been a few edits to my answer since I originally posted it. The original version had a bit at the top about it being against the spirit of the question but intended to amuse. It's not a practical solution at all.

Re 1: I'm not sure there was any particular inspiration. I've worked on quantum computing since 2004, so I guess I spend a lot of time thinking how to make computers in weird ways. Plus I spent college working at a number of ISPs doing tech support, so I used to be reasonably up on networks.

Re 2: Yep. In my field there is this idea of quantum sneaker-net. This is far more useful than my answer and potentially solves a major problem in the field I work in, but is also totally off the wall.

The basic idea is that you can make an extremely high speed low latency network for quantum communication by shipping giant refridgerators around the world on ships (using a quantum effect and the existing [non-quantum] internet). The paper is here: https://www.nature.com/articles/srep36163

Re 3: Somebody here pointed out pingfs which seems to take the idea to a whole new extreme, but reading the website it seems like it was already being worked on in 2011. I doubt anyone implemented my answer in any way. It was somewhat intentionally impractical.